# inclusion-exclusion question check solution help

#### jvignacio

Hey guys, can someone please check my solution if its correct for the following question thankyou!!!

QUESTION:

Use inclusion-exclusion to find the number of solutions in non-negative integers to:

$$\displaystyle X_1 + X_2 + X_3 + X_4 + X_5 = 73$$ with the conditions $$\displaystyle X_2 \leq 30$$, $$\displaystyle 5 \leq X_3 \leq 40$$, $$\displaystyle X_4 \geq 7$$.

MY SOLUTION

STEP 1: Solutions for greater than conditions.

Condition 1: $$\displaystyle 5 \leq X_3 = X_3 \geq 5$$
Condition 2: $$\displaystyle X_4 \geq 7$$

Let

$$\displaystyle X_3 = Y_3 + 5$$
$$\displaystyle X_4 = Y_4 +7$$

Therefore:

$$\displaystyle X_1 + X_2 + (Y_3 + 5) + (Y_4 + 7) + X_5 = 73$$
$$\displaystyle X_1 + X_2 + Y_3 + Y_4 + X_5 = 61$$

So $$\displaystyle \binom{n+k-1}{k} = \binom{5+61-1}{61} = + \binom{65}{61}$$

STEP 2: Solutions for less than conditions.

Condition 1: $$\displaystyle X_2 \leq 30 \Rightarrow X_2 \geq 31$$
Condition 2: $$\displaystyle 5 \leq X_3 \Rightarrow X_3 = Y_3 + 5$$

$$\displaystyle \Rightarrow 5 \leq Y_3 + 5 \leq 40$$ $$\displaystyle = 0 \leq Y_3 \leq 35 =$$ $$\displaystyle Y_3 \leq 35 \Rightarrow Y_3 \geq 36$$

Condition 1:

Let

$$\displaystyle X_2 = Y_2 + 31$$

Therefore:

$$\displaystyle X_1 + (Y_2 + 31) + X_3 + X_4 + X_5 = 61$$
$$\displaystyle X_1 + Y_2 + X_3 + X_4 + X_5 = 30$$

So $$\displaystyle \binom{n+k-1}{k} = \binom{5+30-1}{30} = - \binom{34}{30}$$

Condition 2:

Let

$$\displaystyle X_3 = Y_3 + 36$$

Therefore:

$$\displaystyle X_1 + X_2 + (Y_3 + 36) + X_4 + X_5 = 61$$
$$\displaystyle X_1 + X_2 + Y_3 + X_4 + X_5 = 25$$

So $$\displaystyle \binom{n+k-1}{k} = \binom{5+25-1}{25} = - \binom{29}{25}$$

Now we need to find solutions for

$$\displaystyle Y_2, Y_3 \Rightarrow Y_2 + 31, Y_3 + 36$$

Therefore:

$$\displaystyle X_1 + (Y_2 + 31) + (Y_3 + 36) + X_4 + X_5 = 61$$
$$\displaystyle X_1 + Y_2 + Y_3 + X_4 + X_5 = -6$$

NO SOLUTIONS SINCE RHS IS A NEGATIVE INTEGER.......

therefore final solutions:

$$\displaystyle \binom{65}{61} -(\binom{34}{30} + \binom{29}{25})$$

Last edited:

#### awkward

MHF Hall of Honor
That looks right to me.

• jvignacio