How many notations are there in nonnegative integer to the e

How many notations are there in non-negative integer to the equation x_1 + x_2 + x_3 + x_4 = 17? How many solutions are there if each x_1, in positive?

Solution

For non negative solutions,

We have 17 that can be distributed into 4 variables which can be 0.

Consider a string of 0s of length 17.

00000000000000000

Our problem can be reduced to finding 3 places in this string of 0s and assigning value to each variable depending on the location of 1.

For example,

for 00010001000001000000, x1 = 3, x2 = 3 x3 = 5, x4 = 6.

Hence, think of it as if you have a string of 20 characters, 3 of which are 1s.

So, there are ²⁰C₃ ways to do that. Hence, the number of solutions = ²⁰C_3.

If each of the number are positive integers, then each of them is atleast 1.

Therefore, x1>=1, x2>=1, x3>=1 and x4>=1.

Now this is similar to the above problem except that consider that the amount you need to distribute is 17-4 = 13 now since each variable is atleast 1.

Therefore, we need to create a string of 13+3=16 where 3 are 1s and 13 are 0s.

Number of ways to do that = ¹⁶C₃. Hence, the number of positive integer solutions possible = ¹⁶C₃