Question

In how many ways can you select $5$ letters from the letters in the English alphabet?

• A. ${}^{26}{C}_5$
• B. ${}^{26}{C}_6$
• C. ${}^{26}{C}_{20}$
• D. ${}^{26}{C}_{21}$

Answer 1

Answer: (A), (D)

${}^{26}{C}_{21}$

The total number of letters in the English alphabet are $26$.

We need to select $5$ letters from $26$ letters.

$5$ letters can be selected from $26$ letters in ${}^{26}{C}_5$ ways.

Also, as we know that
${}^n{C}_r={}^n{C}_{n-r}=\frac{n!}{r!(n-r)!}$

Putting the value of $n$ as $26$ and the value of $r$ as $5$:
${}^{26}{C}_5={}^{26}{C}_{26-5}$
${}^{26}{C}_5={}^{26}{C}_{21}$


$5$ letters can be selected from $26$ letters in ${}^{26}{C}_5$ or ${}^{26}{C}_{21}$ ways.

Therefore, option (a) and (d.) are the correct answers.