Question

a typical platoon consists of 20 soldiers. if there are 27 soldiers available to create a platoon, how many different platoons are possible? platoons

Explanation:

Step1: Identify combination formula

The problem is about combinations. The formula for combinations is $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n$ is the total number of items, and $r$ is the number of items to be chosen. Here, $n = 27$ (total soldiers) and $r=20$ (soldiers in a platoon).

Step2: Calculate factorial values

We know that $n!=n\times(n - 1)\times\cdots\times1$. So, $C(27,20)=C(27,27 - 20)=C(27,7)$ (since $C(n,r)=C(n,n - r)$). Then $C(27,7)=\frac{27!}{7!(27 - 7)!}=\frac{27!}{7!×20!}=\frac{27\times26\times25\times24\times23\times22\times21}{7\times6\times5\times4\times3\times2\times1}$.
$27\times26\times25\times24\times23\times22\times21 = 2960104320$ and $7\times6\times5\times4\times3\times2\times1=5040$.

Step3: Compute the combination value

$\frac{2960104320}{5040}=588003$.

Answer:

$588003$