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.

If atleast one cap has to be distributed and the caps have to be arranged such that any box can have a maximum of one cap only, in how many ways can arrange the caps among 5 boxes?

• A. 1
• B. 16
• C. 1545
• D. none of these

Answer 1

The correct option is C 1545

I cap can be selected in ${}^5{C}_1$ ways.
Now this selected cap can be arranged in 5 boxes in ${}^5{P}_1$ ways.
Again 2 caps can be selected in ${}^5{C}_2$ ways.
Now, these selected caps can be arranged in 5 boxes in ${}^5{P}_2$
ways and so on.
Hence the required number of ways
${}^5{C}_1{\times }^5{P}_1{+}^5{C}_2{\times }^5{P}_2{+}^5{C}_3{+}^5{C}_4{\times }^5{P}_4{+}^5{C}_5{\times }^5{P}_5=25+200+600+600+120=1545$