Question

Three letters are randomly selected from the $26$ capital letters of the English alphabet. What is the probability that letters A will not be included in the choice?

• A. $\frac{1}{12}$
• B. $\frac{23}{26}$
• C. $\frac{12}{13}$
• D. $\frac{25}{26}$

Answer 1

The correct option is B

$\frac{23}{26}$

Explanation for the correct option:

Finding the probability that letters A will not be included in the choice:

There are a total of $26$ ways to choose three letters from the alphabet is${=}^{26}{C}_3$

There are $25$ letters left if A is not chosen, hence the number of methods to choose three letters without include $A$ is limited${=}^{25}{\mathrm{C}}_3$

The Resultant probability$=\frac{{}^{25}{\mathrm{C}}_3}{{}^{26}{\mathrm{C}}_3}$

$$\begin{gathered} =\frac{{\displaystyle \frac{25!}{3!\left(22\right)!}}}{{\displaystyle \frac{26!}{3!\left(24\right)!}}}\left[{\because }^n{C}_r=\frac{n!}{r!\left(n-r\right)!}\right] \\ =\frac{25\times 24\times 23}{26\times 25\times 24} \\ =\frac{{\displaystyle 23}}{{\displaystyle 26}} \end{gathered}$$

Hence, the correct option is B.