Question

Evaluate 10c4 and 11p5

Answer 1

To evaluate permutations and combinations, we use the following formulas:

$\begin{gathered} nPr=\frac{n!}{(n-r)!} \\ \\ nCr=\frac{n!}{r!(n-r)!} \end{gathered}$

where n is the total number of objects and r is the number of objects selected from the set.

Following the formulas above, we can solve 10C4 and 11P5.

$\begin{gathered} 10C4=\frac{10!}{4!(10-4)!} \\ \\ 10C4=\frac{10!}{4!\text{ }6!} \\ \\ 10C4=\frac{10\cdot9\cdot8\cdot7\cdot6!}{4\cdot3\cdot2\cdot1\cdot6!} \\ \\ 10C4=10\cdot3\cdot7 \\ \\ 10C4=210 \end{gathered}$$\begin{gathered} 11P5=\frac{11!}{(11-5)!} \\ \\ 11P5=\frac{11!}{6!} \\ \\ 11P5=\frac{11\cdot10\cdot9\cdot8\cdot7\cdot6!}{6!} \\ \\ 11P5=11\cdot10\cdot9\cdot8\cdot7 \\ \\ 11P5=55,440 \end{gathered}$

The answers are 210 and 55,440.