Question

select the correct answer.
what is the number of possible combinations of 14 objects taken 3 at a time?
a. 364
b. 1,001
c. 2,184
d. 24,024

Explanation:

Step1: Recall combination formula

The formula for combinations is \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 14 \) and \( k=3 \).

Step2: Calculate factorials

First, find \( n!=14! = 14\times13\times12\times11\times\cdots\times1 \), \( k!=3! = 3\times2\times1=6 \), and \( (n - k)!=(14 - 3)!=11! = 11\times10\times\cdots\times1 \).

Step3: Substitute into formula

\( C(14,3)=\frac{14!}{3!(14 - 3)!}=\frac{14\times13\times12\times11!}{6\times11!} \). The \( 11! \) terms cancel out.

Step4: Simplify the expression

\( \frac{14\times13\times12}{6}=\frac{14\times13\times2}{1}=14\times26 = 364 \).

Answer:

A. 364