MGMAT Combination Problem

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ritika_bsg: Junior | Next Rank: 30 Posts; Posts: 24; Joined: Wed Sep 01, 2010 1:27 am; Thanked: 1 times; Followed by:1 members

MGMAT Combination Problem

by ritika_bsg » Fri Oct 05, 2012 6:58 am

Each year, a college admissions committee grants a certain number of $10,000 scholarships, $5,000 scholarships, and $1,000 scholarships. The number of scholarships granted at each level does not vary from year to year, and no student can receive more than one scholarship. This year, how many different ways can the committee distribute the scholarships among the pool of 10 applicants?

(1) In total, six scholarships will be granted.

(2) An equal number of scholarships will be granted at each scholarship level.

How do we arrive at the correct answer ?

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Source: Beat The GMAT — Data Sufficiency | Fri Oct 05, 2012 6:58 am

gmat6087: Senior | Next Rank: 100 Posts; Posts: 53; Joined: Sat Oct 29, 2011 3:49 am; Thanked: 2 times; Followed by:1 members

by gmat6087 » Fri Oct 05, 2012 8:19 am

ritika_bsg wrote:Each year, a college admissions committee grants a certain number of $10,000 scholarships, $5,000 scholarships, and $1,000 scholarships. The number of scholarships granted at each level does not vary from year to year, and no student can receive more than one scholarship. This year, how many different ways can the committee distribute the scholarships among the pool of 10 applicants?

(1) In total, six scholarships will be granted.

(2) An equal number of scholarships will be granted at each scholarship level.

How do we arrive at the correct answer ?

In option 1) It tells the total number of scholarships as 6, but we do not know the frequency of each scholarship slab.
It can be
1*10k+ 4*5k+ 1*1k
or
3*10k+ 2*5k+ 1*1k

Since it is a permutation problem we need to know the how many times each scholarship slab is repeated. Not sufficient

2. It tells about the frequency but doesn't give the total possibilities
It can be
2*10k+ 2*5k+ 2*1k
or
3*10k+ 3*5k+ 3*1k
Not sufficient.

Using both solution can be found.
Frequency: 2*10k+ 2*5k+ 2*1k
Total cases:6 out of 10
1st scholarship can be distributed in 10 ways ,2nd in 9 ways so on 6th in 5 ways

=> (10*9*8*7*6*5)/2!*2!*2! answer.

This problem is inline with another simple permutation problem.

Q)In how many ways figures a,a,b,b,c,c can be arranged in a bucket, which has 10 slots.

Hope this helps

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Fri Oct 05, 2012 10:50 am

ritika_bsg wrote:Each year, a college admissions committee grants a certain number of $10,000 scholarships, $5,000 scholarships, and $1,000 scholarships. The number of scholarships granted at each level does not vary from year to year, and no student can receive more than one scholarship. This year, how many different ways can the committee distribute the scholarships among the pool of 10 applicants?

(1) In total, six scholarships will be granted.

(2) An equal number of scholarships will be granted at each scholarship level.

How do we arrive at the correct answer ?

This is a PIECES-OF-THE-PUZZLE question.
Don't solve.
Determine whether there is SUFFICIENT information to solve.

Statement 1: In total, six scholarships will be granted.
A piece is missing: Are the scholarships all of the SAME type or of DIFFERENT types?
Without this information, we cannot determine the number of ways to distribute the scholarships.
INSUFFICIENT.

Statement 2: An equal number of scholarships will be granted at each scholarship level.
A piece is missing: How MANY of each type are being granted?
Without this information, we cannot determine the number of ways to distribute the scholarships.
INSUFFICIENT.

Statements 1 and 2 together:
Now we have ALL OF THE PIECES of the puzzle: two of each type of scholarship are being granted.
Thus, the total number of possible distributions can be determined.
SUFFICIENT.

The correct answer is C.

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Sun Oct 07, 2012 8:33 pm

ritika_bsg wrote:Each year, a college admissions committee grants a certain number of $10,000 scholarships, $5,000 scholarships, and $1,000 scholarships. The number of scholarships granted at each level does not vary from year to year, and no student can receive more than one scholarship. This year, how many different ways can the committee distribute the scholarships among the pool of 10 applicants?

(1) In total, six scholarships will be granted.

(2) An equal number of scholarships will be granted at each scholarship level.

How do we arrive at the correct answer ?

Let us assume that number of $10,000 scholarships = A,
Number of $5,000 scholarships = B,
Number of $1,000 scholarships = C, and
Total number of scholarships granted = T
Then T = A + B + C
Number of different ways the committee can distribute scholarships among the pool of 10 applicants = 10CT * (A + B + C)!/(A! * B! * C!) = 10CT * T!/(A! * B! * C!)

Now we need to know the values of A, B, C, and T to answer the question.

(1) In total, six scholarships will be granted implies T = 6, but we don't know A, B, and C; NOT sufficient.

(2) An equal number of scholarships will be granted at each scholarship level. implies A = B = C, but again we don't know A, B, and C; NOT sufficient.

Combining (1) and (2), T = 6, A = B = C
T = A + B + C implies 6 = A + B + C, which implies that A = B = C = 2

So, required number of ways = 10CT * T!/(A! * B! * C!) = 10C6 * 6!/(2! * 2! * 2!); SUFFICIENT.

The correct answer is C.

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Gurome, Inc.
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