Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.
It is given that there are total 6 red balls, 5 white balls and 5 blue balls. Total 9 balls are to be selected in such a way that each selection consists of 3 balls of each colour.
Since 3 balls have to be selected from 6 red balls, thus it can be shown as the number of balls is the combination of 6 red balls taken 3 at a time.
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 6 for n and 3 for r in the above formula.
C 6 3 = 6 ! ( 6 − 3 ) ! 3 ! = 6 ! 3 ! 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 6 3 = 6 × 5 × 4 × 3 ! 3 ! × 3 × 2 × 1 = 6 × 5 × 4 3 × 2 × 1 = 20
The number of ways that the red balls are selected is 20.
In the same way, since 3 white balls have to be selected from 5white balls, thus it can be shown as the number of balls is the combination of 5white balls taken 3 at a time.
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r !
Substitute 5 for n and 3 for r in the above formula.
C 5 3 = 5 ! ( 5 − 3 ) ! 3 ! = 5 ! 2 ! 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 5 3 = 5 × 4 × 3 ! 3 ! × 2 × 1 = 5 × 4 2 = 10
The number of ways that the white balls are selected is 10.
Also, since 3 blue balls have to be selected from 5 blue balls, thus it can be shown as the number of balls is the combination of 5 blue balls taken 3 at a time.
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 5 for n and 3 for r in the above formula.
C 5 3 = 5 ! ( 5 − 3 ) ! 3 ! = 5 ! 2 ! 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 5 3 = 5 × 4 × 3 ! 3 ! 2 × 1 = 5 × 4 2 = 10
The number of ways that the blue balls are selected is 10.
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 total number of ways is m × n .
The number of ways of selecting 9 balls is,
20 × 10 × 10 = 2000
Thus, 9 balls are selected in 2000 ways with 3 balls of each colour.
It is given that there are total 6 red balls, 5 white balls and 5 blue balls. Total 9 balls are to be selected in such a way that each selection consists of 3 balls of each colour.
Since 3 balls have to be selected from 6 red balls, thus it can be shown as the number of balls is the combination of 6 red balls taken 3 at a time.
The formula for the combination is defined as,
Substitute 6 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,
The number of ways that the red balls are selected is 20.
In the same way, since 3 white balls have to be selected from 5white balls, thus it can be shown as the number of balls is the combination of 5white balls taken 3 at a time.
The formula for the combination is defined as,
Substitute 5 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,
The number of ways that the white balls are selected is 10.
Also, since 3 blue balls have to be selected from 5 blue balls, thus it can be shown as the number of balls is the combination of 5 blue balls taken 3 at a time.
The formula for the combination is defined as,
Substitute 5 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,
The number of ways that the blue balls are selected is 10.
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 total number of ways is
The number of ways of selecting 9 balls is,
Thus, 9 balls are selected in 2000 ways with 3 balls of each colour.
