A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.
It is given that a bag contains 5 black and 6 red balls.
Since 2 black balls are to be chosen then the number of ways of selecting them are the combination of 5 black balls taken 2 at a time.
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r !
Substitute 5 for n and 2 for r in the above formula.
C 5 2 = 5 ! ( 5 − 2 ) ! 2 ! = 5 ! 3 ! 2 !
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 is written as,
C 5 2 = 5 × 4 × 3 ! 3 ! 2 × 1 = 5 × 4 2 = 10
The number of ways that the black balls are selected is 10.
Since 3 red balls are to be chosen then the number of ways of selecting them are the combination of 6 black 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 is 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.
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 that 2 black and 3 red balls are selected is,
10 × 20 = 200
Thus, required number of ways is 200.
It is given that a bag contains 5 black and 6 red balls.
Since 2 black balls are to be chosen then the number of ways of selecting them are the combination of 5 black balls taken 2 at a time.
The formula for the combination is defined as,
Substitute 5 for n and 2 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 is written as,
The number of ways that the black balls are selected is 10.
Since 3 red balls are to be chosen then the number of ways of selecting them are the combination of 6 black 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 is written as,
The number of ways that the red balls are selected is 20.
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 that 2 black and 3 red balls are selected is,
Thus, required number of ways is 200.
