A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (i) exactly 3 girls? (ii) atleast 3 girls? (iii) atmost 3 girls?
It is given that a committee of 7 has to be formed from 9 boys and 4 girls.
(i)
Since exactly 3 girls have to be selected from 4 girls, thus it can be shown as the number of girls is the combination of 4 girls taken 3 at a time.
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 4 for n and 3 for r in the above formula.
C 4 3 = 4 ! ( 4 − 3 ) ! 3 ! = 4 ! 1 ! 3 !
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
n ! = n ( n − 1 ) ! n ! = n ( n − 1 ) ( n − 2 ) ! [ n ≥ 2 ] ⋮
The combination can be written as,
C 4 3 = 4 × 3 ! 3 ! 1 ! = 4 1 = 4
Thus, the number of ways that the girls are selected is 4.
Since the committee consists of total 7 members from which 3 are girls then the number of boys will be, 7 − 3 = 4 .Therefore, 4 boys have to be selected from 9 boys, thus it can be shown as the number of boys is the combination of 9 boys taken 4 at a time.
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 9 for n and 4 for r in the above formula.
C 9 4 = 9 ! ( 9 − 4 ) ! 4 ! = 9 ! 5 ! 4 !
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
n ! = n ( n − 1 ) ! n ! = n ( n − 1 ) ( n − 2 ) ! [ n ≥ 2 ] ⋮
The combination can be written as,
C 9 4 = 9 × 8 × 7 × 6 × 5 ! 5 ! 4 × 3 × 2 × 1 = 9 × 8 × 7 × 6 4 × 3 × 2 × 1 = 126
Thus, the number of ways that the boys are selected is 126.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that exactly 3 girls are selected is 4 × 126 = 504 .
(ii)
If at least 3 girls are to be selected, then the condition arises that the girls can be more than 3 but not less than 3. In this case, two cases that arise are as follows:
(a) 3 girls and 4 boys.
(b) 4 girls and 3 boys.
In case (a), 3 girls are chosen from 4 girls, thus the combination can be written as C 4 3 .
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 4 for n and 3 for r in the above formula.
C 4 3 = 4 ! ( 4 − 3 ) ! 3 ! = 4 ! 1 ! 3 !
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
n ! = n ( n − 1 ) ! n ! = n ( n − 1 ) ( n − 2 ) ! [ n ≥ 2 ] ⋮
The combination can be written as,
C 4 3 = 4 × 3 ! 3 ! 1 ! = 4 1 = 4
Thus, the number of ways that the girls are selected is 4.
Also 4 boys are chosen from 9 boys, thus the combination is C 9 4 .
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 9 for n and 4 for r in the above formula.
C 9 4 = 9 ! ( 9 − 4 ) ! 4 ! = 9 ! 5 ! 4 !
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
n ! = n ( n − 1 ) ! n ! = n ( n − 1 ) ( n − 2 ) ! [ n ≥ 2 ] ⋮
The combination can be written as,
C 9 4 = 9 × 8 × 7 × 6 × 5 ! 5 ! 4 × 3 × 2 × 1 = 9 × 8 × 7 × 6 4 × 3 × 2 × 1 = 126
Thus, the number of ways that the boys are selected is 126.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that exactly 3 girls and 4 boys are selected is 4 × 126 = 504 .
In case (b), 4 girls are chosen from 4 girls, thus the combination can be written as C 4 4 .
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 4 for n and 4 for r in the above formula.
C 4 4 = 4 ! ( 4 − 4 ) ! 4 ! = 4 ! 0 ! 4 ! = 1 1 = 1
Thus, the number of ways that the girls are selected is 1.
Also 3 boys are chosen from 9 boys, thus the combination is C 9 3 .
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 9 for n and 3 for r in the above formula.
C 9 3 = 9 ! ( 9 − 3 ) ! 3 ! = 9 ! 6 ! 3 !
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
n ! = n ( n − 1 ) ! n ! = n ( n − 1 ) ( n − 2 ) ! [ n ≥ 2 ] ⋮
The combination can be written as,
C 9 3 = 9 × 8 × 7 × 6 ! 6 ! 3 × 2 × 1 = 9 × 8 × 7 3 × 2 × 1 = 84
Thus, the number of ways that the boys are selected is 84.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that exactly 4 girls and 3 boys are selected is 1 × 84 = 84 .
Thus, the number of ways that at least 3 girls are selected is 504 + 84 = 588 .
(iii)
If at most 3 girls are to be selected, then the condition arises that the girls can be less than 3 but not more than 3. In this case, following cases arises:
(a) 3 girls and 4 boys.
(b) 2 girls and 5 boys.
(c) 1 girl and 6 boys.
(d) No girl and 7 boys.
In case (a), 3 girls are chosen from 4 girls, thus the combination can be written as C 4 3 .
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 4 for n and 3 for r in the above formula.
C 4 3 = 4 ! ( 4 − 3 ) ! 3 ! = 4 ! 1 ! 3 !
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
n ! = n ( n − 1 ) ! n ! = n ( n − 1 ) ( n − 2 ) ! [ n ≥ 2 ] ⋮
The combination can be written as,
C 4 3 = 4 × 3 ! 3 ! 1 ! = 4 1 = 4
Thus, the number of ways that the girls are selected is 4.
Also 4 boys are chosen from 9 boys, thus the combination is C 9 4 .
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r !
Substitute 9 for n and 4 for r in the above formula.
C 9 4 = 9 ! ( 9 − 4 ) ! 4 ! = 9 ! 5 ! 4 !
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
n ! = n ( n − 1 ) ! n ! = n ( n − 1 ) ( n − 2 ) ! [ n ≥ 2 ] ⋮
The combination can be written as,
C 9 4 = 9 × 8 × 7 × 6 × 5 ! 5 ! 4 × 3 × 2 × 1 = 9 × 8 × 7 × 6 4 × 3 × 2 × 1 = 126
Thus, the number of ways that the boys are selected is 126.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that 3 girls and 4 boys are selected is 4 × 126 = 504 .
In case (b), 2 girls are chosen from 4 girls, thus the combination can be written as C 4 2 .
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 4 for n and 2 for r in the above formula.
C 4 2 = 4 ! ( 4 − 2 ) ! 2 ! = 4 ! 2 ! 2 ! = 4 × 3 × 2 ! 2 ! 2 × 1 = 6
Thus, the number of ways that the girls are selected is 6.
Also 5 boys are chosen from 9 boys, thus the combination is C 9 5 .
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 9 for n and 5 for r in the above formula.
C 9 5 = 9 ! ( 9 − 5 ) ! 5 ! = 9 ! 4 ! 5 ! = 9 × 8 × 7 × 6 × 5 ! 5 ! 4 × 3 × 2 × 1 = 126
Thus, the number of ways that the boys are selected is 126.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that 2 girls and 5 boys are selected is 6 × 126 = 756 .
In case (c), 1 girl is chosen from 4 girls, thus the combination can be written as C 4 1 .
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 4 for n and 1 for r in the above formula.
C 4 1 = 4 ! ( 4 − 1 ) ! 1 ! = 4 ! 3 ! 1 ! = 4 × 3 ! 3 ! 1 = 4
Thus, the number of ways that the girls are selected is 4.
Also 6 boys are chosen from 9 boys, thus the combination is C 9 6 .
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 9 for n and 6 for r in the above formula.
C 9 6 = 9 ! ( 9 − 6 ) ! 6 ! = 9 ! 3 ! 6 ! = 9 × 8 × 7 × 6 ! 6 ! 3 × 2 × 1 = 84
Thus, the number of ways that the boys are selected is 84.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that 1 girl and 6 boys are selected is 4 × 84 = 336 .
In case (d), no girl is chosen from 4 girls, thus the combination can be written as C 4 0 .
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 4 for n and 0 for r in the above formula.
C 4 0 = 4 ! ( 4 − 0 ) ! 0 ! = 4 ! 4 ! 0 ! = 1 1 = 1
Thus, the number of ways that the girls are selected is 1.
Also 7 boys are chosen from 9 boys, thus the combination is C 9 7 .
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 9 for n and 7 for r in the above formula.
C 9 7 = 9 ! ( 9 − 7 ) ! 7 ! = 9 ! 2 ! 7 ! = 9 × 8 × 7 ! 7 ! 2 × 1 = 36
Thus, the number of ways that the boys are selected is 36.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that no girl and 7 boys are selected is 1 × 36 = 36 .
Thus, the number of ways that at most 3 girls are selected is, 504 + 756 + 336 + 36 = 1632
It is given that a committee of 7 has to be formed from 9 boys and 4 girls.
(i)
Since exactly 3 girls have to be selected from 4 girls, thus it can be shown as the number of girls is the combination of 4 girls taken 3 at a time.
The formula for the combination is defined as,
Substitute 4 for n and 3 for r in the above formula.
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The combination can be written as,
Thus, the number of ways that the girls are selected is 4.
Since the committee consists of total 7 members from which 3 are girls then the number of boys will be,
The formula for the combination is defined as,
Substitute 9 for n and 4 for r in the above formula.
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The combination can be written as,
Thus, the number of ways that the boys are selected is 126.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that exactly 3 girls are selected is
(ii)
If at least 3 girls are to be selected, then the condition arises that the girls can be more than 3 but not less than 3. In this case, two cases that arise are as follows:
(a) 3 girls and 4 boys.
(b) 4 girls and 3 boys.
In case (a), 3 girls are chosen from 4 girls, thus the combination can be written as
The formula for the combination is defined as,
Substitute 4 for n and 3 for r in the above formula.
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The combination can be written as,
Thus, the number of ways that the girls are selected is 4.
Also 4 boys are chosen from 9 boys, thus the combination is
The formula for the combination is defined as,
Substitute 9 for n and 4 for r in the above formula.
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The combination can be written as,
Thus, the number of ways that the boys are selected is 126.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that exactly 3 girls and 4 boys are selected is
In case (b), 4 girls are chosen from 4 girls, thus the combination can be written as
The formula for the combination is defined as,
Substitute 4 for n and 4 for r in the above formula.
Thus, the number of ways that the girls are selected is 1.
Also 3 boys are chosen from 9 boys, thus the combination is
The formula for the combination is defined as,
Substitute 9 for n and 3 for r in the above formula.
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The combination can be written as,
Thus, the number of ways that the boys are selected is 84.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that exactly 4 girls and 3 boys are selected is
Thus, the number of ways that at least 3 girls are selected is
(iii)
If at most 3 girls are to be selected, then the condition arises that the girls can be less than 3 but not more than 3. In this case, following cases arises:
(a) 3 girls and 4 boys.
(b) 2 girls and 5 boys.
(c) 1 girl and 6 boys.
(d) No girl and 7 boys.
In case (a), 3 girls are chosen from 4 girls, thus the combination can be written as
The formula for the combination is defined as,
Substitute 4 for n and 3 for r in the above formula.
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The combination can be written as,
Thus, the number of ways that the girls are selected is 4.
Also 4 boys are chosen from 9 boys, thus the combination is
The formula for the combination is defined as,
Substitute 9 for n and 4 for r in the above formula.
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The combination can be written as,
Thus, the number of ways that the boys are selected is 126.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that 3 girls and 4 boys are selected is
In case (b), 2 girls are chosen from 4 girls, thus the combination can be written as
The formula for the combination is defined as,
Substitute 4 for n and 2 for r in the above formula.
Thus, the number of ways that the girls are selected is 6.
Also 5 boys are chosen from 9 boys, thus the combination is
The formula for the combination is defined as,
Substitute 9 for n and 5 for r in the above formula.
Thus, the number of ways that the boys are selected is 126.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that 2 girls and 5 boys are selected is
In case (c), 1 girl is chosen from 4 girls, thus the combination can be written as
The formula for the combination is defined as,
Substitute 4 for n and 1 for r in the above formula.
Thus, the number of ways that the girls are selected is 4.
Also 6 boys are chosen from 9 boys, thus the combination is
The formula for the combination is defined as,
Substitute 9 for n and 6 for r in the above formula.
Thus, the number of ways that the boys are selected is 84.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that 1 girl and 6 boys are selected is
In case (d), no girl is chosen from 4 girls, thus the combination can be written as
The formula for the combination is defined as,
Substitute 4 for n and 0 for r in the above formula.
Thus, the number of ways that the girls are selected is 1.
Also 7 boys are chosen from 9 boys, thus the combination is
The formula for the combination is defined as,
Substitute 9 for n and 7 for r in the above formula.
Thus, the number of ways that the boys are selected is 36.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the number of ways that no girl and 7 boys are selected is
Thus, the number of ways that at most 3 girls are selected is,
