How many solutions are there to $x_{1} + x_{2} + x_{3} + x_{4} = 15$

Substituting $$\begin{cases} y_1=x_1-1\\ y_2=x_2-2\\ y_3=x_3-7\\ y_4=x_4-2 \end{cases}$$ your problem becomes equivalent to finding $y_1,\ldots,y_4$ satisfying $$\begin{split}0&\leq y_1\leq 3,\\0&\leq y_2\leq 3,\\0&\leq y_3,\\0&\leq y_4\end{split}$$ and $$y_1+y_2+y_3+y_4=3.$$ The solution to this are $$y=(y_1,\ldots,y_4)=(3,0,0,0)$$ (and other permutations, making up a total of $4$), $$y=(2,1,0,0)$$ (and other permutations, making up a total of $4!/2!=12$), and $$y=(1,1,1,0)$$ (and other permutations, making up $4$). The number of solutions then is $$4+12+4=20.$$


Write $y_1=x_1-1$, $y_2=x_2-2$, $y_3-7$ and $y_4-2$, then $y_i \geq 0$, $y_1,y_2\leq 3$ and $$y_1+y_2+y_3+y_4 = 3$$

But then $y_3$ and $y_4$ can not be 4 or more. So $y_1,y_2\leq 3$ is not actual restriction here. By stars and bars method we have $${6\choose 3}=20$$ solution.

Tags:

Combinatorics

Related

Difficult Laplace Transform Type of Integral How can I find the value of this [pathological] function? Is there a standard name for this relation property : " aRb --> there is no c different from b such that aRc "? How can i Prove that the gray area is the same as white area? Why probability of picking a random prime is 0? Show $ f(x) = \sum_{n=1}^{\infty} \frac{nx}{n^3 + x^3}$ ,$\ g(x) = \sum_{n=1}^{\infty} \frac{x^4n}{(n^3 + x^3)^2}$ are bounded on $[0, \infty)$. Spanning forests of bipartite graphs and distinct row/column sums of binary matrices Let $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$. Prove that $x_{n+1} < x_{n}$ Can we show a sum of symmetrical cosine values is zero by using roots of unity? Simplifying sigma of sigma of sigma Calculate $\int e^{2x}(\cos x)^3 dx$ Integration problem in $\lim_{n\to\infty}\sum_{i=1}^n{{\ln(n+5i)}\over n}-\ln n$

Recent Posts