Question

Following questions contains statements given in two columns. which have to be matched. The statements in Colum -I are tabelled as AB, B, C and D while the statements is Column -II are labelled a sp, q, r and s. Any given statement in Column-I can have correct matching with One statemetn in Column-II

5 balls are to be placed in 3 boxes. Each box can hold all the 5 balls. Number of ways in which the balls can be placed so that no box remains empty. If.

$\begin{array}{cccc}& Column-I& & Column-II\\ \left(A\right)& ballsareidenticalbutboxesaredifferent& \left(p\right)& 2\\ \left(B\right)& ballsaredifferentbutboxesxareidentical& \left(q\right)& 25\\ \left(C\right)& ballsaswellasboxesareidentical& \left(r\right)& 50\\ \left(D\right)& Ballsaswellasboxesareidenticalbutboxesarekeptinarow& \left(s\right)& 6\end{array}$

Answer 1

B) Balls are different but boxes are identical
The arrangement in boxes doesn't matters boxes are identical
$5=1+2+2\to$ No. of ways$=5C_1\times 4C_2{\times }^2{C}_2=5\times 6\times 1=30$
$5=1+3+1\to No.ofways=5C_1{\times }^4{C}_3{\times }^1{C}_1=5\times 4\times 1=20$
$\therefore$ Total number of ways = 30+20=50

C) Boxes as well as balls are identical
There are only 2 arrangements
$\frac{1}{1},\frac{2}{3},\frac{2}{1}$

D) Only option left is 25