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  • A school committee of 12 members is to be formed from a group of 19 males and 16 females. If m represents the number of ways the committee can be formed with at least 7 males, and n represents the

A school committee of 12 members is to be formed from a group of 19 males and 16 females. If m represents the number of ways the committee can be formed with at least 7 males, and n represents the number of ways the committee can be formed with at least 8 females. Fine the value of m and n?

 

Option: 1

64,567


Option: 2

83,879


Option: 3

38,506


Option: 4

54,658


Answers (1)

best_answer

To find the value of m and n, we need to calculate the number of ways the committee can be formed with at least 7 males and at least 8 females.

For m, we will calculate the number of ways the committee can be formed with 7, 8, 9, 10, 11, or 12 males.

m = Number of ways with 7 males + Number of ways with 8 males + Number of ways with 9 males + Number of ways with 10 males + Number of ways with 11 males + Number of ways with 12 males

To calculate the number of ways with a specific number of males, we use the combination formula. The number of ways to choose k males from a group of n males is given by the combination formula:

\mathrm{C(n, k)=n ! /(k ! \times (n-k) !)}

Similarly, for n, we will calculate the number of ways the committee can be formed with 8, 9, 10, 11, or 12 females.

n = Number of ways with 8 females + Number of ways with 9 females + Number of ways with 10 females + Number of ways with 11 females + Number of ways with 12 females

Let's calculate the value of n by evaluating each term:

Number of ways with 8 females  \mathrm{=C(16,8)=16 ! /(8 ! \times(16-8) !)=12870}
Number of ways with 9 females = \mathrm{C(16,9)=16 ! /(9 ! \times(16-9) !)=11440}
Number of ways with 10 females = \mathrm{C(16,10)=16 ! /(10 ! \times(16-10) !)=8008}
Number of ways with 11 females =\mathrm{C(16,11)=16 ! /(11 ! \times(16-11) !)=4368}
Number of ways with 12 females = \mathrm{C(16,12)=16 ! /(12 ! \times(16-12) !)=1820}
 

Therefore, the value of n is:
n = 12870 + 11440 + 8008 + 4368 + 1820 = 38506


 

Posted by

HARSH KANKARIA

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