Question

There are 5 different caps $c_1,c_2,c_3,c_{4}and{c}_5$ and 5 different boxes $B_1,B_2,B_3,B_{4}and{B}_5$. The capacity of each box is sufficient to accomodate all the 5 caps.

In how many arrangements does $B_1$ have cap $C_1$?

• A. 5!
• B. $5^4$
• C. ${}^5{P}_4$
• D. none of these

Answer 1

The correct option is B $5^4$

Required number of arrangements $=5\times 5\times 5\times 5=625$
Since we fix cap $C_1$ in box $B_1$ then we arrange the remaining 4 caps in any of the 5 boxes
Alternatively: Total number of arrangements $=5^5$
Number of arrangements in which $B_1$ have cap $C_1=\frac{5^5}{5}=5^4$