How many solutions are there to $x_1 + x_2 + ... + x_5 = 21$?

Question

How many solutions are there to the equation

$x_1 + x_2 + x_3 + x_4 + x_5 = 21$

where $x_1, i = 1,2,3,4,5$ is a nonnegative integer such that $0 ≤ x_1 ≤ 3, 1 ≤ x_2 < 4$, and $x_3 ≥ 15$?

I have correctly completed the previous parts of the question but am having trouble with this particular restriction.

The total number of solutions to the equation is $C(25,21) = 12650$.

I have tried to do each restriction individually:

For $0 ≤ x_1 ≤ 3$, I solved this by subtracting the number of solutions when $x_1 ≥ 4$ from the total number of solutions. This is $C(25,21) - C(21,17) = 5985$.

For $1 ≤ x_2 < 4$, I have $C(24,20) - C(21,17) = 5985$.

For $x_3 ≥ 15$, I have $C(10,6) = 210$.

However, the restriction is when these are all together. I am not sure how to proceed. The answer is $106$.

Answer 1

The system:

$\begin{cases} x_1+x_2+x_3+x_4+x_5=21\\ 0\leq x_1\leq 3\\ 1\leq x_2<4\\ 15\leq x_3\\ 0\leq x_4\\ 0\leq x_5\end{cases}$

can be rewritten via a change of variables as:

$\begin{cases} y_1+y_2+y_3+y_4+y_5=5\\ 0\leq y_1\color{blue}{\leq 3}\\ 0\leq y_2\color{blue}{\leq 2}\\ 0\leq y_3\\ 0\leq y_4\\ 0\leq y_5\end{cases}$

Do you see how?

Let $x_1=y_1, x_2=y_2+1, x_3=y_3+15$ and $x_4=y_4$ and $x_5=y_5$. Each of the conditions translate directly as above, including $y_1+y_2+y_3+y_4+y_5=x_1+x_2-1+x_3-15+x_4+x_5=21-1-15=5$

Now, approach via inclusion-exclusion on the first two terms. If we were to ignore the upper bounds, how many solutions are there?

$\binom{5+5-1}{5-1}=\binom{9}{4}$

How many of those solutions violate the first upper bound condition?

that would be when $y_1>3$ i.e. $4\leq y_1$. Making another change of variable, this would be $z_1+z_2+\dots+z_5=1, 0\leq z_i$ for a total of $\binom{5}{4}=5$

How many of them violated the second upper bound condition?

Approach similarly

How many violate both upper bound conditions simultaneously?

That would be with $x_1\geq 4, x_2\geq 4, x_3\geq 15$ implying that $x_1+x_2+x_3+x_4+x_5\geq x_1+x_2+x_3\geq 23$ and couldn't equal $21$, so none violate both.

Letting $|S|$ be the amount which ignore both upper bound conditions, $|A_1|$ be the amount which violate the first upper bound condition, $|A_2|$ be the amount which violate the second upper bound condition, we have the number of solutions which violate none of the upper bound conditions as:

$$|S|-|A_1|-|A_2|+|A_1\cap A_2|$$

$126 - 5 - 15 + 0 = 106$

Answer 2

However, the restriction is when these are all together. I am not sure how to proceed.

Utilise the Principle of Inclusion and Exclusion and the Twelve Fold Way's "Stars and Bars" technique.

The total number of natural number solutions is: $\binom{21+5-1}{21}$.   This is the count of ways to place $21$ balls into $5$ boxes.   The "boxes" are the variables, and the value is incremented by $+1$ for each "ball"; so "five balls in box 1" corresponds to the event $\{x_1=5\}$.   You grok that okay?

Let us label the restrictions as $A:=\{0≤x_1 ≤3\}$, $B:=\{1≤x_2 <4\}$ and $C:=\{x_3 ≥15\}$

Now restriction $C$ alone is easy to count:   When at least $15$ balls must be placed in box three; we need only count the ways to distribute the remaining $6$ balls among all five boxes.

$$\lvert{C}\rvert=\binom{6+5-1}{6}$$

The other restrictions are more easily countable through their complements; though we'll split that for $B$ into two disjoint subsets (you'll see why soon).

Let $A'=\{x_2\geq 4\}, B'_1=\{x_2=0\}, B'_2=\{x_2\geq 4\}$

Then, noting that $B'_1\cap B'_2=\emptyset$, the Principle of Inclusion and Exclusion says "hello": $$\begin{align}\lvert{A\cap B\cap C}\rvert ~=~&\lvert{C}\rvert-\lvert{(A'\cup B_1'\cup B_2')\cap C}\rvert\\[1ex]~=~&\lvert{C}\rvert-\lvert{A'\cap C}\rvert-\lvert{B_1'\cap C}\rvert-\lvert{B_2'\cap C}\rvert+\lvert{A'\cap B_1'\cap C}\rvert+\lvert{A'\cap B_2'\cap C}\rvert\end{align}$$

To measure $\lvert{A'\cap C}\rvert$ : When at least $15$ balls are placed in box three and at least $4$ balls in box one; we must count the ways to distribute the remaining $2$ balls among all five boxes.

$$\lvert{A'\cap C}\rvert = \binom{2+5-1}{2}$$

To measure $\lvert{B_1'\cap C}\rvert$ : when $15$ are placed in box three and box two is kept empty; we must count the ways to distribute the remaining $6$ balls among the remaining four boxes.

$$\lvert{B'_1}\rvert = \binom{6+4-1}{6}$$

And so forth...

Answer 3

An approach using generating functions:

In order to count up the number of solutions that sum to $n$, we can look at the coefficient of $x^n$ in the generating function $f(x)$:

$$[x^{n}]f(x) = [x^{n}]\left(\sum_{n=0}^{3}x^n\right)\left(\sum_{n=1}^{3}x^n\right)\left(\sum_{n=15}^{\infty}x^n\right)\left(\sum_{n=0}^{\infty}x^n\right)\left(\sum_{n=0}^{\infty}x^n\right)$$

The $[x^n]$ operator refers to the coefficient of $x^n$.

Factor out $x$'s so you can shift the indices down to $n=0$ for each sum:

$$[x^{n}]f(x) = [x^{n}]\left(\sum_{n=0}^{3}x^n\right)x\left(\sum_{n=0}^{2}x^n\right)x^{15}\left(\sum_{n=0}^{\infty}x^n\right)\left(\sum_{n=0}^{\infty}x^n\right)\left(\sum_{n=0}^{\infty}x^n\right)$$

Use the identities $\displaystyle \sum_{n=0}^{\infty}x^n = \frac{1}{1-x}$ and $\displaystyle \sum_{n=0}^{k}x^n = \frac{1-x^{k+1}}{1-x}$ to convert each piece:

$$[x^{n}]f(x) = [x^{n}]\left(\frac{1-x^4}{1-x}\right)x\left(\frac{1-x^3}{1-x}\right)x^{15}\left(\frac{1}{1-x}\right)^3$$

Simplify:

$$[x^{n}]f(x) = [x^{n}]\left(\frac{x^{23}-x^{20}-x^{19}+x^{16}}{(1-x)^5}\right)$$

Factor out $\dfrac{1}{(1-x)^5}$ and use the identity $\displaystyle \frac{1}{(1-x)^{m+1}} = \sum_{n=0}^{\infty}\binom{n+m}{m}x^n$:

$$[x^{n}]f(x) = [x^{n}]\left(x^{23}-x^{20}-x^{19}+x^{16}\right)\sum_{n=0}^{\infty}\binom{n+4}{4}x^n$$

Distribute:

$$[x^{n}]f(x) = [x^{n}]\left(\sum_{n=0}^{\infty}\binom{n+4}{4}x^{n+23}-\sum_{n=0}^{\infty}\binom{n+4}{4}x^{n+20}-\sum_{n=0}^{\infty}\binom{n+4}{4}x^{n+19}+\sum_{n=0}^{\infty}\binom{n+4}{4}x^{n+16}\right)$$

Shift indices:

$$[x^{n}]f(x) = [x^{n}]\left(\sum_{n=23}^{\infty}\binom{n-19}{4}x^{n}-\sum_{n=20}^{\infty}\binom{n-16}{4}x^{n}-\sum_{n=19}^{\infty}\binom{n-15}{4}x^{n}+\sum_{n=16}^{\infty}\binom{n-12}{4}x^{n}\right)$$

If we look at the lower bounds of the indices, the smallest $x^n$ that we could get the coefficient for is $x^{16}$. This is because $16$ is the smallest possible value of $n$ under the $x_i$ constraints, which occurs when $x_1 = x_4 = x_5 = 0$, $x_2=1$, and $x_3=15$. The coefficients for all $n<16$ are simply $0$.

Now we can invoke the coefficient operator:

$$[x^n]f(x) = \binom{n-19}{4}-\binom{n-16}{4}-\binom{n-15}{4}+\binom{n-12}{4}$$

Set $n=21$ to get the number of solutions to the original problem:

$$[x^{21}]f(x) = 106$$

Side note: If you use $n<19$ in the expression above, you'll start to see negative binomial coefficients. For those, you need to use the identity $\binom{-n}{k} = (-1)^k\binom{n+k-1}{k}$

Answer 4

Let's compute it directly

let $y_3 = x_3-15$ and $y_2 = x_2-1$ and replace $x_2$ and $x_3$ in the main sum.

$x_1+y_2+x_3+x_4+y_5 = 21-1-15 = 5$

Now, let's apply the theorem 2 of stars and bars with $n = 5 , k = 5$ to compute the total of comb. including the not desired $x_1>=4$ and $x_2 >=4$ : $\tbinom{n + k - 1}{n}$ = $\tbinom{5 + 5 - 1}{5}$ = whole = $\tbinom{9}{5} = 126 $

Let's now apply $x_1 < 4$ and $y_2 < 3$

$x_1$ != 5 : only (5,0,0,0,0) , then 1 comb

$x_1$ != 4 : (4,?,?,?,?), then stars and bars again with $n=1 , k=4$ : $\tbinom 4 1 = 4$ comb

$y_2$ != 5 : only (0,5,0,0,0) , then 1 comb

$y_2$ != 4 : all the (?,4,?,?,?) , S&B with $n=1 , k=4$ : $\tbinom 4 1 = 4$ comb

$y_2$ != 3 : all the (?,3,?,?,?) , S&B with $n=2 , k=4$ : $\tbinom 5 1 = 10$ comb

then we get exclusions = $1 + 1 + 4 + 4 + 10 = 20 $ comb.

Result = whole - exclusions = $126 - 20 = 106$ combinations.