The chorus has 35 members. Which expression represents the number of ways a group of 6 members can be chosen to do a special performance?

A. [tex]$35 \cdot 6$[/tex]

B. [tex]$\frac{35!}{6! \cdot 29!}$[/tex]

C. [tex]$\frac{35!}{29!}$[/tex]



Answer :

To determine the number of ways to choose 6 members from a group of 35, we use the concept of combinations. A combination is a selection of items without considering the order. The mathematical formula for a combination is given by:

[tex]\[ C(n, k) = \frac{n!}{k!(n-k)!} \][/tex]

Where [tex]\( n \)[/tex] is the total number of items, and [tex]\( k \)[/tex] is the number of items to choose. In this case, [tex]\( n = 35 \)[/tex] and [tex]\( k = 6 \)[/tex]. Therefore, the number of ways to choose 6 members from 35 is:

[tex]\[ C(35, 6) = \frac{35!}{6!(35-6)!} = \frac{35!}{6! \cdot 29!} \][/tex]

Now, let's match this with the given options:

1. [tex]\( 35 \cdot 6 \)[/tex]

This expression represents a simple multiplication and does not account for the combination formula. Hence, it is incorrect.

2. [tex]\( \frac{35!}{6!29!} \)[/tex]

This expression correctly represents the formula for combinations. Therefore, it is the correct answer.

3. [tex]\( \frac{35!}{29!} \)[/tex]

This expression represents a permutation where the order matters and is not the same as a combination with [tex]\( k = 6 \)[/tex]. Hence, it is incorrect.

In conclusion, the correct expression that represents the number of ways a group of 6 members can be chosen from 35 members is:

[tex]\[ \boxed{\frac{35!}{6!29!}} \][/tex]