Question

List all the combinations of 5 objects $a,b,c,d$, and $e$taken 3 at a time. What is$C(5,3)$?

Short answer

The list of combinations is:

$abc,abd,abe,acd,ace,ade,bcd,bce,bde,cde.$

The value of the given combination is 10.

Step-by-step solution

Step 1. Given information.

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

The given combinations is $C(5,3)$.

Step 2. Determine the list of combinations.

The list of combinations is:

$abc,abd,abe,acd,ace,ade,bcd,bce,bde,cde.$

Step 3. Determine the value of combination.

We know that,

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

Using the above formula, we get

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