TIFR CSS GS-2017

Question

How many distinct ways are there to split 50 identical coins among three people so that each person gets atleast 5 coins??

Answer 1

Your question can be reduced in the form -: 

$x+y+z=50$

where,

$x\geq 5,y\geq 5,z\geq 5,$

so the above eqaution can be written as 

$\left (p+5 \right )+\left ( q+5 \right )+\left ( r+5 \right )=50$

$p+q+r=35$

It is same as Number of ways of distributing n identical objects among r groups

which is $\binom{35+3-1}{3-1}=\binom{37}{2}$

Comments

Sir I have doubt ..why not answer 37C3 ... in most of the non integral questions solution   n+r-1Cr only we used this formula ...