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Question
There are 4 balls of different colours & 4 boxes of colours same as those of the balls. The number of ways in which the balls, one in each box, could be placed such that exactly no ball go to the box of its own colour is:
There are 4 balls of different colours & 4 boxes of colours same as those of the balls. The number of ways in which the balls, one in each box, could be placed such that exactly no ball go to the box of its own colour is:
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Solution
The correct option is A 9
Total no. of ways in which balls can be placed =4!. No. of ways in which exactly 1 ball goes to box of its own colour =4×2!=8 ways.No. of ways in which exactly 2 balls go to box of their own colour =4C2×1!=6 ways.No. of ways in which exactly 3 balls go to box of their own colour =0 [ ∴ if 3 balls go to box of their colour, further will also automatically go ]No. of ways in which exactly 4 balls go to box of their own colour =1∴ No. of ways in which no ball goes to box of its own colour =24−8−6−0−1 =9 ways.Hence, the answer is 9.
Total no. of ways in which balls can be placed =4!. No. of ways in which exactly 1 ball goes to box of its own colour =4×2!=8 ways.
No. of ways in which exactly 2 balls go to box of their own colour =4C2×1!=6 ways.
No. of ways in which exactly 3 balls go to box of their own colour =0 [ ∴ if 3 balls go to box of their colour, further will also automatically go ]
No. of ways in which exactly 4 balls go to box of their own colour =1
∴ No. of ways in which no ball goes to box of its own colour =24−8−6−0−1
=9 ways.
Hence, the answer is 9.
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