Question

List all the combinations of 5 objects $a,b,c,d,$and $e$taken 2 at a time. What is $C(5,2)$?

Short answer

The list of combinations is:

$ab,ac,ad,ae,bc,bd,be,cd,ce,de.$

The value of the given combination is 10.

Step-by-step solution

Step 1. Given information.

2 objects are taken at a time from 5 objects $a,b,c,d,$and $e$at a time.

The given combinations is $C(5,2)$.

Step 2. Determine the list of combinations.

The list of combinations is:

$ab,ac,ad,ae,bc,bd,be,cd,ce,de.$

Step 3. Determine the value of combination.

We know that,

$C(n,r)=\frac{n!}{(n-r)!r!}$

Using the above formula, we get

$\begin{array}{rcl}C(5,2)& =& \frac{5!}{(5-2)!2!}\\ & =& \frac{5!}{3!2!}\\ & =& \frac{5\times 4\times 3!}{3!\times 2\times 1}\\ & =& 10\end{array}$