Question

There are three bags $B_1,B_2,\text{and}{B}_3$ containing $2$ red & $3$ white, $5$ red & $5$ white, $3$ red & $2$ white balls respectively. A ball is drawn from bag $B_1$ and placed in $B_2$. Then a ball is drawn from bag $B_2$ and placed in $B_3$. Then a ball is drawn from bag $B_3$. The number of ways in which this process can be completed, if same colour balls are used in the first and the second transfers is (assuming all balls are different)

• A. $30$
• B. $180$
• C. $150$
• D. $25$

Answer 1

The correct option is B $180$

Case I:
If red colour ball is drawn in the first and the second transfer, then the number of ways $={}^2{C}_1\times {}^6{C}_1\times {}^6{C}_1=72$

Case II:
If white colour ball is drawn in the first and the second transfer, then the number of ways $={}^3{C}_1\times {}^6{C}_1\times {}^6{C}_1=108$

$\therefore$ Total the number of ways$=72+108=180$