combinatorics

Question

THERE ARE 10 PRIZES AND 4 STUDENTS. WE WANT ONLY TWO STUDENTS TO GET THE PRIZES. HOW MANY WAYS ARE THERE??

Comments

@sushmita,thats because a single grade can be given to more than 1 person,like

A can have A+ grade and B can also have A+ grade.so,each of the 4 students have 5 choices,thats why 5⁴

but a prize can be given to a single person,i.e first prize cannot be shared.so,each prize has  4 options.

am i clear??

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Thanx  Akriti. It's clear now. From where to clear all combinatorics? Its sometimes so confusing

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sushmita,i too get confused in combinatorics.i too struggle in PnC.i think solving as much questions  as u can can only help u.

Answer 1

Here's a simple approach:

Ways of selecting pair of 2 students=  4C₂ =  6

Ways of distributing 10 prizes among 2 students= 2¹⁰ = 1024

Total ways till now = 6 * 1024  = 6144 but there are cases that we want to remove like student A getting all the 10 prizes. As A appears in 3 pairs i.e (A,B), (A,C), (A,D). Hence 3 cases for student A. Total such cases for 4 students = 3 * 4= 12

Hence 6144 - 12 = 6132 ways

Answer 2

C(4,2)*10*9 ?

Answer 3

the solution is-

4C2 * 2^10 -3*4C3*1^10+4C2*0^10=6132