Question

If there are no restrictions on where the digits and letters are placed, how many $8$-place license plates consisting of $5$ letters and $3$ digits are possible if no repetitions of letters or digits are allowed? What if the $3$ digits must be consecutive?

Short answer

Possible no. of license plates if no repetitions of letters or digits are allowed$318,269,952,000$.

Possible no. of license plates if the $3$ digits must be consecutive$34,100,352,000$.

Step-by-step solution

Step 1. Find possible no. of license plates if no repetitions of letters or digits are allowed.

There are $26$letters and $10$digits.

No. of ways in which $3$positions for digits can be selected are $\left(\begin{array}{c}8\\ 3\end{array}\right)=\frac{8!}{3!5!}=56$

Out of the localid="1649226801190" $26$letters,

the first letter can be selected in $26$ways.

the second letter can be selected in $25$ways.

the third letter can be selected in $24$ways.

the fourth letter can be selected in $23$ways.

the fifth letter can be selected in $22$ways.

Out of the localid="1649226808269" $10$digits,

the first letter can be selected in $10$ways.

the first letter can be selected in $9$ways.

the first letter can be selected in $8$ways.

Therefore, the possible no. of license plates if no repetitions of letters or digits are allowed islocalid="1649227324212" $56\times 26\times 25\times 24\times 23\times 22\times 10\times 9\times 8=318,269,952,000$

Step 2. Find the possible no. of license plates if the 3 digits must be consecutive

If $3$digits must be consecutive then the possible positions for digits are $=3!=6$

Therefore, the possible no. of license plates if the $3$ digits must be consecutive are$6\times 26\times 25\times 24\times 23\times 22\times 10\times 9\times 8=34,100,352,000$