Question

How many different selections of 6 books can be made from 10 different books?

• A. ${}^6{C}_{10}$
• B. $\frac{7!}{4!}$
• C. $5040$
• D. (a) and (b) above.

Answer 1

The correct option is B $\frac{7!}{4!}$

We know that number of ways of selecting "k" things out of "n" things at a time is equal to ${}^n{C}_k=\frac{n!}{(n-k)!.k!}.$

So, according to question number of ways of selecting $6$ books out of $10$ different books is equal to ${}^{10}{C}_6.$
$\Rightarrow {}^{10}{C}_6=\frac{10!}{(10-6)!.6!}=\frac{10!}{4!.6!}$

$\Rightarrow {}^{10}{C}_6=\frac{10\times 9\times 8\times 7\times 6!}{4!.6!}=\frac{5040}{24}=210$

Also, we know that $7!=5040$ and $4!=24$.

Dividing both we will get the desired answer as $\frac{7!}{4!}=\frac{5040}{24}=210$