Question

A variable name for a certain computer language must be either an alphabet or an alphabet followed by a decimal digit. The total number of different variable names that can exist in that language is equal to.

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Hint: The formula for determining the number of possible arrangements by selecting only a few objects from a set with no repetition is expressed in the following way:
${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
$N =$the total number of elements in a set
$K =$ The number of selected objects (the order of the objects is not important)
$! = factorial$

Complete step by step solution:
The variable name consists only of a letter, then that letter can be chosen in 26 ways from the alphabets.
If the code is created using a letter followed by a digit, then code can be created in
${}^{26}{C_1} \times {}^{10}{C_1}$
These two events are independent
Now,
Above combination can be done in only one way.
$1! = 1$
$\Rightarrow \dfrac{{26!}}{{1!(26 - 1)!}} \times \dfrac{{10!}}{{1!(10 - 1)!}} \times 1$ $\left[ {{}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}} \right]$
$\Rightarrow \dfrac{{26.25!}}{{25!}} \times \dfrac{{10.9!}}{{9!}}$ $\left[ \begin{gathered} n = 26 \\ r = 1 \\ \end{gathered} \right]$
$\Rightarrow 26 \times 10$
$\Rightarrow 260$
Hence
Total number of different variable names that can exist in that language is
$= 26 + 260$
$= 286$

Note:
$10! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10$
So, it can be writing as
$10! = 9!\, \times 10$
1)(Remember)$1! = 1$
2) In the above question, it is fixed followed by a decimal digit, So the rotation can be done by only 1 way.