Question

there are 7 women and 6 men signed up to join a swing dance class. In how many ways can the instructor choose 4 of the people to join if 1 or fewer must be men?

Answer 1

SOLUTION

Step 1: write out the parameters given

$\begin{gathered} 7\text{women} \\ 6\text{men} \end{gathered}$

The instructor is to choose 4 people in a swing dance

Hence we have a combination

Step2: Write out the combination expression

Since one men of fewer is to be choosen then we gave

$(7C3\times6C1)+(7C4+6C0)$

Srep3; Simplify the expression above

$\begin{gathered} \text{ Recall that} \\ nCr=\frac{n!}{(n-r)!r!} \\ 7C3=\frac{7!}{(7-3)!3!}=\frac{7!}{4!3!}=35 \\ 6C1=\frac{6!^{}}{(6-1)!1!}=\frac{6!}{5!1!^{}}=6 \\ \end{gathered}$

Then

$\begin{gathered} 7C4=\frac{7!}{(7-4)!4!}=\frac{7!}{3!4!}=35 \\ 7C0=\frac{7!}{(7-0)!0!}=\frac{7!}{7!0!}=1 \\ \text{where 0!=1} \end{gathered}$

Step4: Substitute the value into the expression in step 2, we obtain

$\begin{gathered} (7C3\times6C1)+(7C4+6C0) \\ (35\times6)+(35\times1) \\ 210+35=245 \end{gathered}$

Therefore the instructor can make the choice in 245 ways