Question

There are six members of a jazz club who are listed below. A concert director can choose some or all of them to be part of a large show. She must choose at least one of them. Fiona, Gary, Han, Joe, Kate, Kyle In how many different ways can the director choose the members for the show? (The order the members are chosen doesn't matter.)

Answer 1

Answer:  63

Step-by-step explanation:

Given : Number of  members of a jazz club = 6

When order doesn't matter , then the number of combinations to select r things out of n is given by :-

$C(n;r)=\dfrac{n!}{r!(n-r)!}$

Now, the number of ways to select at least one of them :-

$^6C_1+^6C_2+^6C_3+^6C_4+^6C_5+^6C_6\\\\=6+\dfrac{6!}{2!4!}+\dfrac{6!}{3!3!}+\dfrac{6!}{2!4!}+6+1\ \ [ \because\ ^nC_0=^nC_n=1]\\\\=6+15+20+15+6+1\\\\=63$

Hence, the number of ways to choose the members for the show = 63