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# In how many ways can 16 different gifts be divided among fou

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In how many ways can 16 different gifts be divided among fou [#permalink]  10 Oct 2017, 20:25
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60% (01:45) correct 40% (01:31) wrong based on 5 sessions
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. $$4^{16}$$

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Re: In how many ways can 16 different gifts be divided among fou [#permalink]  10 Oct 2017, 21:08
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.

Here,

16 gifts are distributed among 4 children such that each receives exactly 4 gifts can be arranged in = 16C4

Now, we are left with 12 gifts which can be arranged in = 12C4

Now, we are left with 8 gifts that can be arranged in = 8C4

then we are left with 4 gifts which can be arranged in= 4C4

Total no. of ways = 16C4 * 12C4 * 8C4 * 4C4

= $$\frac{(16!)}{(4!)^4}$$
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Director
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Re: In how many ways can 16 different gifts be divided among fou [#permalink]  11 Oct 2017, 05:54
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Bunuel wrote:
In how many ways can 16 different gifts be divided among four children such that each child receives exactly four gifts?

One more approach is by using partition method

Therefore The number of possible partition of 16 gifts into 4 groups (4 children) of 4 gifts

or $$P$$ (4,4,4,4) = $$\frac{16!}{(4!*4!*4!*4!)}$$ = $$\frac{16!}{(4!)^4}$$
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Re: In how many ways can 16 different gifts be divided among fou   [#permalink] 11 Oct 2017, 05:54
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