Mini Concept: Multinomial Theorem
Some Trickery of Multinomial Theorem
x+y+z=n the number of non negative solutions of this equation will be
if we extend this to r variables then the formula becomes (n+r-1)C(r-1)
if we remove 0, means we need only positive integral solutions to the equation then we get formula as (n-1)C(r-1)
Lets take up one example.
On the occasion of Diwali, PAPA CHIPS is offering one of five prizes with every packet( the prize is inside the packet). the prizes include a pen, pencil, a CD, a movie ticket and a small game. Banta Singh is a fan of PAPA chips and he keeps buying the chips, what is the probability that Banta Singh gets all the five prizes by buying 12 packets of PAPA chips
Chuck the story, the question is there are 5 variables and we need the solutions to the equation
a+b+c+d+e=12 ( non negative)
and a+b+c+d+e=12 ( positive integral)
The first case comes as every packet has a prize, and those 5 are the only kinds of prizes.
Second comes from that we need each kind of prize.
so the answer is 11C4/16C4=11!12!/(7!16!)=126.96.36.199/188.8.131.52= 33/182
Lets take another example
If the sum of 101 distinct terms in arithmetic progression is zero , in how many ways can three of these terms be selected such that their sum is zero?
it is obvious that the middle term is zero
so the terms are
-50D, -49D,….,-D, 0, D, ….49D, 50D
now the sum of 3 numbers to be zero
Case 1) if we pick 0, then we have to pick one positive and one negative, which must be equal except for the sign . so 50 ways
case 2) we leave 0 and pick two positive and one negative
z can vary from 1 to 50
we need positive solutions to the equation
which comes 0C1+1C1+2C1…+49C1
add this it will come to 50C2
case 3 it will be same as case 2
we get 50C2
hence total is 2.50C2+50=2500