GATE2011 MN: GA-59

Question

In how many ways $3$ scholarships can be awarded to $4$ applicants, when each applicant can receive any number of scholarships?

• A. $4$ 
• B. $12$ 
• C. $64$ 
• D. $81$

Answer 1

One scholarship can be awarded to $4$ applicants in $4$ ways. Three scholarship can be awarded to

$4$ applicants in $4^{3} = 64$ ways.

(All scholarship are independent, any student can get any no of scholarships between 0 and 3)

Correct Answer: $C$

Comments

Why it is not n+r-1cr format? 6c3=20 ways

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Because the scholarships and applicants are both distinguishable. This is not explicitly given, but it is understood.

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Why the option is not $4^{4}$ ??

I mean 3 scholarships available and each student can have 4 choices = {0,1,2,3}

Here 0 means the student haven’t got the scholarship and 3 means it have all the 3 scholarships.

So,   it should be   =   4 * 4 * 4 * 4   =  $4^{4}$.     (one 4 for each student)

           

Answer 2

Added this answer for extra clarity.

Take 1^st scholarship, it can be given to any of the 4 applicants. So 4 ways.

As it is given that,

each applicant can receive any number of scholarships

An applicant can have any number of scholarships from 0 to all 3. Suppose an applicant gets all 3 scholarships, the rest of the applicants will get no scholarships.

So it is possible that the applicant having the 1^st scholarship may also get the 2^nd scholarship.

Take 2^nd scholarship, it can also be given to any of the 4 applicants. So 4 ways.

Take 3^rd scholarship, it can also be given to any of the 4 applicants. So 4 ways.

Total number of ways 3 scholarships can be given to 4 applicants = 4 * 4 * 4 = 64 ways.