+0

# Counting

0
119
3

How many solutions are there to the equation
u + v + w + x + y + z = 18
where u, v, w, x, y, and z are nonnegative integers, and x is at most 10?

Mar 5, 2023

#1
+195
0

Let's consider two cases: one where x is less than or equal to 10, and one where x is greater than 10.

Case 1: x is less than or equal to 10.
If x is at most 10, then there are 11 possible values for x (0, 1, 2, ..., 10). For each value of x, we can count the number of nonnegative integer solutions to the equation u + v + w + y + z = 18 - x using stars and bars. There are 5 bars and 18 - x stars, so the number of solutions is (18 - x + 5 C 5). Therefore, the total number of solutions when x is at most 10 is:

(0 + 5 C 5) + (1 + 5 C 5) + (2 + 5 C 5) + ... + (10 + 5 C 5)
= (5 C 5) + (6 C 5) + (7 C 5) + ... + (15 C 5)
= 2002

Case 2: x is greater than 10.
If x is greater than 10, then we can subtract 11 from x to get a new variable x' that is at most 7. Then we can rewrite the equation as u + v + w + x' + y + z = 7, where u, v, w, y, and z are nonnegative integers. Using stars and bars, the number of solutions to this equation is (7 + 5 C 5) = 792.

Therefore, the total number of solutions is 2002 + 792 = 2794.

So there are 2794 nonnegative integer solutions to the equation u + v + w + x + y + z = 18, where x is at most 10.

Mar 5, 2023
#2
0

a=listfor(n, 1, 11, (11+n) nCr 4);print a, "=", sum a

(495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315) = 32857 - total number of solutions.

Mar 5, 2023
#3
+195
0

The code you provided calculates the values of the binomial coefficients (n choose 4) for n from 1 to 11 using the Python syntax for binomial coefficients, `(11+n) nCr 4`, and stores them in a list called `a`.

The print statement then outputs the values of `a` and their sum. The sum of `a`, which is calculated correctly by the code, is 32857.

However, the statement that this is the total number of solutions is not correct without additional context. It is unclear what problem these binomial coefficients are related to, so it is difficult to determine whether the total number of solutions is indeed 32857.

Justingavriel1233  Mar 6, 2023