In how many ways can Rs. 16 be divided into 4 person when none of them get less than Rs.3
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Solution :
Required number of ways
= coefficient of x16 in (x3+x4+x5+.......x7)4
= coefficient of x16 in x12(1+x+x2+.......x4)4
= coefficient of x16 in x12(1−x5)4(1−x)−4
= coefficient of x4 in (1−x5)4(1−x)−4
= coefficient of x4 in (1−4x5+.....)
[1+4x+....+(r+1)(r+2)(r+3)3!x]
= (4+1)(4+2)(4+3)3! = 35
Aliter: Remaining 4 rupees can be distributed in 4+4−1C4−1 i.e., 35 ways
This can be mathematically formulated as
w+x+y+z=16 and each of them is greater or equal to 3 and we need to find number of all possible values of x w x y and z.
The number of non negative integral solution to the equation: (x1)+(x2)+(x3)+.....(xr)=n is given as (n+r-1)C(r-1)
Using above formula we will make a simple assumption
a=w-3, b=x-3, c=y-3 and d=z-3
So now the equation reduces
a+b+c+d=4
Solution is 7C3 that is 35
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