chaitanya.mehrotra wrote:Gordon buys 5 dolls for his 5 nieces. The gifts include two identical Sun-and-Fun
beach dolls, one Elegant Eddie dress-up doll, one G.!. Josie army doll, and one Tulip
Troll doll. If the youngest niece does not want the G.!. Josie doll, in how many different
ways can he give the gifts?
The solution below assumes that every niece is to receive exactly 1 gift.
Good = Total - Bad.
Total:
Number of ways to arrange 5 distinct elements = 5! = 120.
But the 5 elements in the problem above are not distinct; 2 of the gifts are identical, decreasing the number of unique arrangements.
To account for the smaller number of unique arrangements, we must divide by the number of ways to arrange the 2 identical elements:
5!/2! = 60.
Bad:
In a bad arrangement, the youngest niece receives the GI Josie doll.
Since there is only 1 choice for the youngest niece -- in a bad arrangement, she must receive the GI Josie Doll -- we need only count the number of ways to arrange the 4 remaining gifts:
4!/2! = 12.
Good = 60-12 = 48.
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