First each child gets at least one gift, so the first 6 gifts are distributed to 6 children (1 each) - 1 way
Now you have 4 gifts remaining and 6 people who can take these gifts.
xxxx |||||
x be the gifts and | represent people (6 people so 5 slots), they can be arranged in 9!/5!4! ways (or) 9C5 ways
The direct formula is the one you mentioned, pretty straightforward, you get 9C5
Gifts
This topic has expert replies
Source: Beat The GMAT — Problem Solving |
-
shankar.ashwin
- Legendary Member
- Posts: 966
- Joined: Sat Jan 02, 2010 8:06 am
- Thanked: 230 times
- Followed by:21 members
-
shankar.ashwin
- Legendary Member
- Posts: 966
- Joined: Sat Jan 02, 2010 8:06 am
- Thanked: 230 times
- Followed by:21 members
The formula you mentioned is ONLY applicable when 'n' things are to be distributed to 'r' people without any restrictions, i.e 1 person can get all 'n' and others get 0 and so on.
We are given a condition that each child receives at least 1 gift here, so the formula cannot be directly applied in this case.
Always make sure you apply formulas correctly when using them, IMO a formulaic approach to P&C often gives misleading answers for the test makers frame questions such as this, where you could go wrong when you use the formula directly. A more logical approach always gets you close to the answer, even for a educated guess.
Once we recognize the problem tests us on (n+r-1)C(r-1), we should ensure we the formula is applicable to our scenario. Hence we initially distribute the first 6 gifts and then use the formula.
A more general way of approaching P&C problems is to always account for *any given conditions* in the problem before using generic counting methods/slot method or even for formulas.
We are given a condition that each child receives at least 1 gift here, so the formula cannot be directly applied in this case.
Always make sure you apply formulas correctly when using them, IMO a formulaic approach to P&C often gives misleading answers for the test makers frame questions such as this, where you could go wrong when you use the formula directly. A more logical approach always gets you close to the answer, even for a educated guess.
Once we recognize the problem tests us on (n+r-1)C(r-1), we should ensure we the formula is applicable to our scenario. Hence we initially distribute the first 6 gifts and then use the formula.
A more general way of approaching P&C problems is to always account for *any given conditions* in the problem before using generic counting methods/slot method or even for formulas.
kakz wrote:@ashwin
Shouldn't the formula be (n-1)C(r-1) in this case? I mean, if we do not consider that each child has first been given 1 gift each at the beginning
-
knight247
- Legendary Member
- Posts: 504
- Joined: Tue Apr 19, 2011 1:40 pm
- Thanked: 114 times
- Followed by:11 members
@Ashwin
I think u slightly misunderstood the question. Kakz said the formula is (n-1)C(r-1) and not
(n+r-1)C(r-1). Try this problem by doing (n-1)C(r-1) which is the formula for when each person gets
atleast 1 and the remaining can be distributed in any way. You'll get the answer 9C5
(n+r-1)C(r-1) is, like u said, the formula which includes distributions where one person gets all etc. Basically, it includes all the restricted possibilities.
Try using the (n-1)C(r-1) on the following problem. U'll see what I'm talking about
https://www.beatthegmat.com/combination-t41362.html
P.S. I've seen a quite similar funda on the Arun Sharma book on CAT Quant. Haven't u seen this funda there?
I think u slightly misunderstood the question. Kakz said the formula is (n-1)C(r-1) and not
(n+r-1)C(r-1). Try this problem by doing (n-1)C(r-1) which is the formula for when each person gets
atleast 1 and the remaining can be distributed in any way. You'll get the answer 9C5
(n+r-1)C(r-1) is, like u said, the formula which includes distributions where one person gets all etc. Basically, it includes all the restricted possibilities.
Try using the (n-1)C(r-1) on the following problem. U'll see what I'm talking about
https://www.beatthegmat.com/combination-t41362.html
P.S. I've seen a quite similar funda on the Arun Sharma book on CAT Quant. Haven't u seen this funda there?
Last edited by knight247 on Thu Jan 05, 2012 10:33 am, edited 1 time in total.
-
shankar.ashwin
- Legendary Member
- Posts: 966
- Joined: Sat Jan 02, 2010 8:06 am
- Thanked: 230 times
- Followed by:21 members
Hey sorry guys... I wasn't really aware of that formula. Guess its applicable for this problem here.
Somehow never had a formulaic approach for P&C, and at least for the GMAT you don't need to know complex P&C rules.
In my opinion this is quite an advanced problem and you would see it unless you're scoring high in quants.
Somehow never had a formulaic approach for P&C, and at least for the GMAT you don't need to know complex P&C rules.
In my opinion this is quite an advanced problem and you would see it unless you're scoring high in quants.
