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]
[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]
