Question

Find the total number of distinct vehicle numbers that can be formed using two letters followed by two numbers. Letters need to be distinct .
A. 60000
B. 65000
C. 70000
D. 75000

Answer 1

Hint: Find the number of ways two distinct letters can be selected. Also find the number of ways two numbers followed after letters are selected . Multiply to get the desired number of ways.

Complete step-by-step answer:
Out of 26 alphabets two distinct letters can be chosen in $^{26}{P_2}$ ways.
Coming to the numbers part, there are 10 ways (any number from 0 to 9 can be chosen) to choose the first digit and similarly another 10 ways to choose the second digit.
Hence there are totally 10×10=100 ways to choose numbers .
Combined with letters there are $^{26}{P_2} \times 100 = \dfrac{{26!}}{{\left( {26 - 2} \right)!}} \times 1000 = 26 \times 25 \times 1000 = 65000$ ways to choose the vehicle numbers.
(since $^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$)
So option B. is correct.

Note: In such kinds of questions, the formula and concept of permutation should be recalled to select the numbers or letters. Note that permutation is an ordered combination which means that in permutation we care about our order of the selection.