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How many ways can it be arranged on a shelf? [#permalink]
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There are 4 copies of 5 different books. In how many ways can they be arranged on a shelf?A) 20!/4!B) 20!/5(4!)C) 20!/(4!)^5D) 20!
E) 5!
Re: How many ways can it be arranged on a shelf? [#permalink]
Alchemist1320 wrote:
There are 4 copies of 5 different books. In how many ways can they be arranged on a shelf?A) 20!/4!B) 20!/5(4!)C) 20!/(4!)^5D) 20!
E) 5!
Let A, B, C, D, and E represent the 5 different booksSo, we want to arrange the following 20 letters: AAAABBBBCCCCDDDDEEEE-------ASIDE---------------------------------------------When we want to arrange a group of items in which some of the items are identical, we can use something called the MISSISSIPPI rule.
It goes like this:If there are n objects where A of them are alike, another B of them are alike, another C of them are alike, and so on, then the total number of possible arrangements = n!/[(A!)(B!)(C!)....]
So, for example, we can calculate the number of arrangements of the letters in MISSISSIPPI as follows:There are 11 letters in total
There are 4 identical I's
There are 4 identical S's
There are 2 identical P's
So, the total number of possible arrangements = 11!/[(4!)(4!)(2!)]----------ONTO THE QUESTION--------------------------GIVEN: AAAABBBBCCCCDDDDEEEE
There are 20 letters in total
There are 4 identical A's
There are 4 identical B's
There are 4 identical C's
There are 4 identical D's
There are 4 identical E's
So, the total number of possible arrangements = 20!/[(4!)(4!)(4!)(4!)(4!)]
= 20!/[(4!)^5]Answer: CCheers,Brent _________________
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Re: How many ways can it be arranged on a shelf? [#permalink]
20!/((4!)^5)Answer - C _________________
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Re: How many ways can it be arranged on a shelf? [#permalink]
Alchemist1320 wrote:
There are 4 copies of 5 different books. In how many ways can they be arranged on a shelf?A) 20!/4!B) 20!/5(4!)C) 20!/(4!)^5D) 20!
E) 5!
formula : The number of ways in which MN different items can be divided equally into M groups, each containing N objects and the order of the groups is important is = (mn)!/(n!)^m20!/(4!)^5= C
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Re: How many ways can it be arranged on a shelf? [#permalink]
Could someone explain me the rationale behind this formula ?
Re: How many ways can it be arranged on a shelf? [#permalink]
Loki2612 wrote:
Could someone explain me the rationale behind this formula ?
Result is option C = 20!/(4!)^5
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Re: How many ways can it be arranged on a shelf? [#permalink]
Look it as number of ways of arranging AAAA BBBB CCCC DDDD EEEE books where A,B,C,D,E repeat 4 times.
hence 20!/ 4!^5
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Re: How many ways can it be arranged on a shelf? [#permalink]
20! / ((4!)^5)this division is done to avoid repetitions.Lets say we have to figure out number of arrangements for A,B1,B2. (where B1=B2) . total arrangements for 3 letters is 3!.A-B1-B2A-B2-B1 - duplicate as B1=B2B1-A-B2B2-A-B1 - duplicate as B1=B2B1-B2-AB2-B1-A - duplicate as B1=B2so to avoid duplicates we need to divide the total arrangements/ (number of similar items)! = 3!/2!
Loki2612 wrote:
Could someone explain me the rationale behind this formula ?
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Re: How many ways can it be arranged on a shelf? [#permalink]
sudhir18n wrote:
Alchemist1320 wrote:
There are 4 copies of 5 different books. In how many ways can they be arranged on a shelf?A) 20!/4!B) 20!/5(4!)C) 20!/(4!)^5D) 20!
E) 5!
formula : The number of ways in which MN different items can be divided equally into M groups, each containing N objects and the order of the groups is important is = (mn)!/(n!)^m20!/(4!)^5= C
thnx for the formula. please tell me what is the formula if order is not important _________________
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Re: How many ways can it be arranged on a shelf? [#permalink]
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Re: How many ways can it be arranged on a shelf? [#permalink]
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Re: How many ways can it be arranged on a shelf? [#permalink]
11 Jul 2022, 02:18