Chapter 13.2, Problem 55AYU

Senate Committees The U.S. Senate has 100 members. Suppose that it is desired to place each senator on exactly 1 of 7 possible committees. The first committee has 22 members, the second has 13, the third has 10, the fourth has 5, the fifth has 16, and the sixth and seventh have 17 apiece. In how many ways can these committees be formed?

To determine

To find: In how many ways can these committee be formed?

Answer to Problem 55AYU

$\approx 1.157*{10}^{76}$

Explanation of Solution

Given:

The U.S. Senate has 100 members. Suppose that it is desired to place each senator on exactly 1 of 7 possible committees. The first committee has 22 members, the second has 13, the third has 10, the fourth has 5, the fifth has 16, and the sixth and seventh have 17 apiece.

Formula used:

$C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$

Calculation:

Number of senators available for the first committee $=100$.

Hence, Number of ways to form first committee is $C\left(100,22\right)$.

Number of senators available for the second committee $=100-22=78$.

Hence, Number of ways to form second committee is $C\left(78,13\right)$.

Number of senators available for the third committee $=78-13=65$.

Hence, Number of ways to form third committee is $C\left(65,10\right)$.

Number of senators available for the fourth committee $=65-10=55$.

Hence, Number of ways to form fourth committee is $C\left(55,5\right)$.

Number of senators available for the fifth committee $=55-5=50$.

Hence, Number of ways to form fifth committee is $C\left(50,16\right)$.

Number of senators available for the sixth committee $=50-16=34$.

Hence, Number of ways to form sixth committee is $C\left(34,17\right)$.

Number of senators available for the seventh committee $=34-17=17$.

Hence, Number of ways to form seventh committee is $C\left(17,17\right)$.

The total number of committees

$=C\left(100,22\right)*C\left(78,13\right)*C\left(65,10\right)*C\left(55,5\right)*C\left(50,16\right)*C\left(34,17\right)*C\left(17,17\right)$

$=\frac{100!}{22!78!}*\frac{78!}{13!65!}*\frac{65!}{10!*55!}*\frac{55!}{50!*5!}*\frac{50!}{16!*34!}*\frac{34!}{17!*17!}*\frac{17!}{17!*1!}$

$\approx 1.157*{10}^{76}$