How many solutions are there to the equation u+v+w+x+y+z

where u, v, w, x, y, and z are nonnegative integers, and x is at most 2?

Capitalist Mar 19, 2023

#1**0 **

We can break down the problem into two cases based on the value of x:

Case 1: x is 0 or 1.

If x is 0 or 1, then there are three possibilities for the value of y: 0, 1, or 2. For each of these values of y, we can break down the problem further based on the value of z:

If z is 0, then we need to find the number of solutions to the equation u+v+w = -x-y, where u, v, and w are nonnegative integers. This is a standard stars and bars problem, and the number of solutions is (3+2-1) choose 2 = 6 choose 2 = 15.

If z is 1, then we need to find the number of solutions to the equation u+v+w = -x-y+z-1. Again, this is a stars and bars problem, and the number of solutions is (3+2-1) choose 2 = 15.

If z is 2, then we need to find the number of solutions to the equation u+v+w = -x-y+z-2. Once again, this is a stars and bars problem, and the number of solutions is (3+2-1) choose 2 = 15.

Therefore, in this case, the total number of solutions is 3*(15+15+15) = 135.

Case 2: x is 2.

If x is 2, then y must be 0. We can break down the problem further based on the value of z:

If z is 0, then we need to find the number of solutions to the equation u+v+w = -x-y = -2, where u, v, and w are nonnegative integers. This is a stars and bars problem, and the number of solutions is (3+2-1) choose 2 = 6 choose 2 = 15.

If z is 1, then we need to find the number of solutions to the equation u+v+w = -x-y+z-1 = -1. This is also a stars and bars problem, and the number of solutions is (3+2-1) choose 2 = 15.

If z is 2, then we need to find the number of solutions to the equation u+v+w = -x-y+z-2 = 0. This is once again a stars and bars problem, and the number of solutions is (3+2-1) choose 2 = 15.

Therefore, in this case, the total number of solutions is 15+15+15 = 45.

Adding up the number of solutions from both cases, we get a total of 135+45 = 180 solutions to the equation.

Guest Mar 19, 2023

#2**0 **

Sorry, that is not the correct answer, thank you so much for giving it a try though!

Capitalist
Mar 19, 2023

#4**+1 **

It is not an equation, it is a sum.

There is an infinite number of integer solutions. Ths smallest is 2 and every integer larger than 2 is also a possible solution.

Melody Mar 20, 2023

#5**0 **

Oh it was ment to equal 8, I'm so sorry I messed up, I'm terribly sorry for the inconvinience.

Once again, I apologize.

Capitalist
Mar 20, 2023