Provide an appropriate response.A club has n (n ? 3) members. They want to choose 3 people from among these n members to form a committee. Prove by induction that the number of possible ways of choosing the 3 people is .

Question

What will be an ideal response?

Answer 1

Answers may vary. Possible answer:
First, we show the statement is true when n = 3.
When n = 3, there is only one way to choose a committee of 3 - all 3 members must be on the committee. When n = 3, 
So P₃ is true and the first condition for the principle of induction is satisfied.
Next, we assume the statement holds for some unspecified natural number k. That is, we assume that P_k is true:
P_k: the number of ways to choose the committee of 3 from k members is  .
On the basis of the assumption that P_k is true, we need to show that P_k+1 is true. 
P_k+1: the number of ways to choose the committee of 3 from k + 1 members is  .
The number of ways to choose the committee of 3 from k + 1 members is the number of ways to choose the committee from the first k members plus the number of ways to choose the committee such that the committee includes the (k + 1)st member and 2 of the first k members.  
Thus, the number of ways to choose the committee of 3 from k + 1 members is 
 +  
=  + 
= 
= 
= 
So P_k+1 is true if P_k is assumed true. Therefore, by the principle of mathematical induction, the number of ways to choose a committee of 3 from n members is  .