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In how many ways can 16 different gifts be divided among fou
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14 Jun 2019, 11:15
1
Bunuel wrote:
In how many ways can 16 different gifts be divided among four children such that each child receives exactly four gifts?
A. \(16^4\)
B. \((4!)^4\)
C. \(\frac{(16!)}{(4!)^4}\)
D. \(\frac{(16!)}{(4!)}\)
E. \(4^{16}\)
Kudos for correct solution.
Let's say the children are named A, B, C, and D
Stage 1: Select 4 gifts to give to child A Since the order in which we select the 4 gifts does not matter, we can use combinations. We can select 4 gifts from 16 gifts in 16C4 ways (= 16!/(4!)(12!)) So, we can complete stage 1 in 16!/(4!)(12!) ways
Stage 2: select 4 gifts to give to child B There are now 12 gifts remaining Since the order in which we select the 4 gifts does not matter, we can use combinations. We can select 4 gifts from 12 gifts in 12C4 ways (= 12!/(4!)(8!)) So, we can complete stage 2 in 12!/(4!)(8!) ways
Stage 3: select 4 gifts to give to child C There are now 8 gifts remaining We can select 4 gifts from 8 gifts in 8C4 ways (= 8!/(4!)(4!)) So, we can complete stage 3 in 8!/(4!)(4!) ways
Stage 4: select 4 gifts to give to child D There are now 4 gifts remaining NOTE: There's only 1 way to select 4 gifts from 4 gifts, but if we want the answer to look like the official answer, let's do the following: We can select 4 gifts from 4 gifts in 4C4 ways (= 4!/4!) So, we can complete stage 4 in 4!/4! ways
By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus distribute all 16 gifts) in [16!/(4!)(12!)][12!/(4!)(8!)][8!/(4!)(4!)][4!/4!] ways
A BUNCH of terms cancel out to give us = 16!/(4!)⁴
Answer: C
Note: the FCP can be used to solve the MAJORITY of counting questions on the GRE. So, be sure to learn it.
Re: In how many ways can 16 different gifts be divided among fou
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04 Jan 2024, 13:04
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Re: In how many ways can 16 different gifts be divided among fou [#permalink]