Question

How many solutions (using only nonnegative integers) are there to the following equation?
x1 + x2 + x3 + x4 + x5 = 33

Answer 1

familiar with stars and bars ?

Answer 2

https://brilliant.org/discussions/thread/stars-and-bars/

Answer 3

\(\large \color{black}{\begin{align}
x+x^2+x^3+x^4+x^5=33\\~\\
\left(x+x^2+x^3+x^4+x^5\right)\pmod 3=33 \pmod 3\\~\\
\left(x+x^2+x^3+x^4+x^5\right)\pmod 3=0\\~\\
\normalsize \text{for x=0}\\~\\
\left(x+x^2+x^3+x^4+x^5\right)\pmod 3=0\\~\\
0\pmod 3\neq 0\\~\\
\normalsize \text{but }\\~\\
0+0^2+0^3+0^4+0^5\neq 33\\~\\
\normalsize \text{for x=1}\\~\\
\left(x+x^2+x^3+x^4+x^5\right)\pmod 3=0\\~\\
2\pmod 3\neq 0\\~\\
\normalsize \text{for x=2}\\~\\
\left(x+x^2+x^3+x^4+x^5\right)\pmod 3=0\\~\\
2\pmod 3\neq 0\\~\\
\normalsize \text{no integer solutions}\\~\\
\end{align}}\)

Answer 4

\(\large \color{black}{\begin{align}

\normalsize \text{basic approach}\\~\\
x^1 + x^2 + x^3 + x^4 + x^5 = 33\\~\\
\normalsize \text{for x=1}\\~\\
x^1 + x^2 + x^3 + x^4 + x^5\\~\\
=5\\~\\
5<33\\~\\
\normalsize \text{for x=2}\\~\\
x^1 + x^2 + x^3 + x^4 + x^5\\~\\
=2^1 + 2^2 + 2^3 +2^4 + 2^5\\~\\
=62\\~\\
33<62\\~\\

\normalsize \text{again no integer solutions}\\~\\
\end{align}}\)

Answer 5

If you meant: \(x_1+x_2+x_3+x_4+x_5=33\) then there will be 66045 answers....

Answer 6

Ex: \(x_1,x_2,x_3,x_4=0,x_5=33\),etc....