# Thread: no of solutions to w+x+y+z=8?

1. ## no of solutions to w+x+y+z=8?

How many solutions in non-negative integers are
there to the equation
w +x+y +z = 8?

and
How many solutions in positive integers?

I think for the first part i am just looking at the number of 8 element multisubsets of a 4 element set e.g (8+4-1) choose 8?
For the second part I am a bit confused,
thanks for any help

2. For the second part think this way.
Put a one in each variable.
That leaves four to put into four places.
$\dbinom{4+4-1}{4}$.

3. Ah thank you that makes sense

4. Hello, hmmmm!

How many solutions in non-negative integers are
there to the equation: . $w +x+y +z \:=\: 8$ ?

Place 8 objects in a row, leaving a space before, after and between them.
. . $
\_\;\circ\;\_\;\circ\;\_\;\circ\;\_\;\circ\;\_\;\c irc\;\_\; \circ\;\_\;\circ\;\_\; \circ\;\_$

Place 3 "dividers" in any of the 9 spaces.

. . $\circ \circ | \circ \circ \circ| \circ \circ |\circ\;\text{ represents }\,2 + 3 + 2 + 1$

. . $\circ \circ || \circ \circ \circ \circ \circ |\circ\;\text{ represents }\,2 + 0 + 5 + 1$

. . $|\circ \circ \circ | \circ \circ \circ \circ \circ|\;\text{ represents }\,0 + 3 + 5 + 0$

. . $\circ \circ \circ \circ \circ \circ |||\circ \circ\;\text{ represents }\,6 + 0 + 0 + 2$

. . $\circ \circ \circ \circ \circ\circ \circ\circ|||\;\text{ represents }\,8 + 0 + 0 + 0$

For each of the three dividers, there are 9 choices for location.

There are: . $9^3 \,=\,729$ non-negative solutions to the equation.

How many solutions in positive integers?

Place 8 objects in a row, leaving a space between them.
. . $\circ\;\_\; \circ\;\_\;\circ\;\_\;\circ\;\_\;\circ\;\_\;\circ\ ;\_\;\circ\;\_\;\circ$

Select three of the 7 spaces in which to insert the dividers.

. . There are: . $_7C_3 \:=\:{7\choose3} \:=\:35$ ways.

There are 35 positive integer solutions to the equation.