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combinations - 5 girls 5 dolls - one constraint

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combinations - 5 girls 5 dolls - one constraint

by nabilqureshi » Wed Jun 02, 2010 7:16 am
Gordon buys 5 dolls for his five nieces. The gifts include two identical TYPE A dolls, one TYPE B doll, one TYPE C doll and one TYPE D doll. If the youngest niece does not want the type D doll, how many different ways can he give the gifts.

I understand the the total number of ways ignoring the constraint is 60 (5!/2!)

To subtract the number of ways that the youngest niece DOES get the type D doll, I use the following logic.

If there are 5 girls and a total of 60 ways, each girl is present in every single one of the 60 combinations.
So looking at all 60 combinations, each girl is getting each type of doll 15 times (60/4).
Not 12 (60/5) because we have already adjusted for the two identical TYPE A dolls (2!)

So how come the answer is not 60 - 15 = 45 (with the constraint)

Thanks!
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by liferocks » Wed Jun 02, 2010 7:23 am
if the youngest nice always get doll D, other dols i.e. AABC can be distributed in other four girls is 4!/2!=12

so IMO ans should be 60-12=48

in yours logic 60 is the total number then each girls getting each type of doll is 60/5=12..so ans is 60-12=48
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by nabilqureshi » Wed Jun 02, 2010 7:38 am
liferocks wrote:if the youngest nice always get doll D, other dols i.e. AABC can be distributed in other four girls is 4!/2!=12

so IMO ans should be 60-12=48

in yours logic 60 is the total number then each girls getting each type of doll is 60/5=12..so ans is 60-12=48
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If the 60 combinations are comprised of each girl getting one of 4 dolls an equal number of times, then each girl must be getting doll D 15 times.

I divide by 4 because the 60 is (5!/2!), meaning we have already adjusted for the two identical dolls, so there are only 4 unique dolls.

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by newyork10r » Sun Oct 17, 2010 8:44 am
nabilqureshi wrote:Gordon buys 5 dolls for his five nieces. The gifts include two identical TYPE A dolls, one TYPE B doll, one TYPE C doll and one TYPE D doll. If the youngest niece does not want the type D doll, how many different ways can he give the gifts.

I understand the the total number of ways ignoring the constraint is 60 (5!/2!)

Thanks!
Hi, I'm just now going through this problem and am having a difficult time understanding the very first step: why is it divided by 2!?
Thanks.
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by GMATGuruNY » Sun Oct 17, 2010 9:32 am
nabilqureshi wrote:Gordon buys 5 dolls for his five nieces. The gifts include two identical TYPE A dolls, one TYPE B doll, one TYPE C doll and one TYPE D doll. If the youngest niece does not want the type D doll, how many different ways can he give the gifts.

I understand the the total number of ways ignoring the constraint is 60 (5!/2!)

To subtract the number of ways that the youngest niece DOES get the type D doll, I use the following logic.

If there are 5 girls and a total of 60 ways, each girl is present in every single one of the 60 combinations.
So looking at all 60 combinations, each girl is getting each type of doll 15 times (60/4).
Not 12 (60/5) because we have already adjusted for the two identical TYPE A dolls (2!)

So how come the answer is not 60 - 15 = 45 (with the constraint)

Thanks!
Number of choices for the youngest niece = 4 (since she can't get the type D doll).

Number of ways to give the remaining 4 dolls = 4!/2! = 12. (We divide by 2! to account for the 2 identical dolls.)

Multiplying, we see that the number of ways to give the dolls = 4*12 = 48.
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