Question

If radio station call letters must begin with either $\mathrm{K}$ or $\mathrm{W}$ and must include either two or three additional letters, How many different possibilities are there?

Answer 1

Step-1: The number of possibilities that $\mathbf{K}$or $\mathbf{W}$ and must include two additional letters:

Given that, Radio Station Call Letters Must Begin With Either $\mathrm{K}$ Or $\mathrm{W}$ And Must Include Either Two Or Three Additional Letters.

The number of possibilities that letter $\mathrm{K}$ or $\mathrm{W}$ and must include two additional letters is obtained below:

From the information, the radio station call letters must begin with the letters either $\mathrm{K}$ or $\mathrm{W}$ and the name should include with $2$ or $3$ additional letters.
The number of English letters is $26$.

$\begin{array}{rcl}\left(\begin{array}{l}\text{The number of}\\ \text{possibilities that}\\ \text{letter}K\text{or}W\text{and}\\ \text{must include two}\\ \text{additional letters}\end{array}\right)& =& \left[\begin{array}{l}\left(\begin{array}{l}\text{Number of}\\ \text{ways to place}\\ \text{letter}W\text{or}K\end{array}\right)\times \left(\begin{array}{l}\text{Number of ways to}\\ \text{select additional of}\\ \text{firstletter from}\\ 26\text{letters}\end{array}\right)\\ \left(\begin{array}{l}\text{Number of ways to}\\ \text{select additional}\\ \text{of secondletter}\\ \text{from 26 letters}\end{array}\right)\end{array}\right]\\ & =& 2\times 26\times 26\\ & =& 2\times {26}^2\end{array}$

The number of possibilities that radio station call letters must begin with either $\mathrm{K}$ or $\mathrm{W}$ and must include either two or three additional letters is obtained below:
Step-2: Find the required number of possibilities:

$\begin{array}{rcl}\left(\begin{array}{l}\text{Number of}\\ \text{possibilities}\\ \text{that radio}\\ \text{station call}\\ \text{letters must}\\ \text{begin with either}\\ K\text{or}W\text{and must}\\ \text{include either two}\\ \text{or three additional letters}\end{array}\right)& =& \left(\begin{array}{l}\text{The number of}\\ \text{possibilities that}\\ \text{letter}K\text{or}W\text{and}\\ \text{must include two}\\ \text{additional letters}\end{array}\right)+\left[\begin{array}{l}\left[\begin{array}{l}\text{Number of}\\ \text{ways to place}\\ \text{letter}W\text{or}K\end{array}\right)\times \\ \left(\begin{array}{l}\text{Number of ways to}\\ \text{select additional of}\\ \text{first letter from}\\ 26\text{letters}\end{array}\right)\times \left(\begin{array}{l}\text{Number of ways to}\\ \text{select additional}\\ \text{of secondletter}\\ \text{from 26 letters}\end{array}\right)\times \\ \left(\begin{array}{l}\text{Number of ways to}\\ \text{select additional}\\ \text{of thirdletter}\\ \text{from 26 letters}\end{array}\right)\end{array}\right]\\ & & \begin{array}{c}=\left(2\times {26}^2\right)+(2\times 26\times 26\times 26)\\ =\left(2\times {26}^2\right)+\left(2\times {26}^3\right)\\ =2\left({26}^2+{26}^3\right)\\ =1,352+35,152\\ =36,504\end{array}\end{array}$

Therefore, the number of possibilities are $\mathbf{36}\mathbf{,}\mathbf{504}$.