Question

How many solutions are there to the equation x₁ + x₂ + x₃ + x₄ + x₅ + x₆ = 25 in which each xi is a non-negative integer and …

Hint (solution for 3 ≤ x₁ ≤ 10):

Let x₁ = y₁ + 3. The condition 3 ≤ x₁ ≤ 10 is equivalent to the condition that 0 ≤ y₁ ≤ 7. The number of solutions to

x₁ + x₂ + x₃ + x₄ + x₅ + x₆ = 25 with 3 ≤ x₁ ≤ 10 is equal to the number of solutions to

y₁ + x₂ + x₃ + x₄ + x₅ + x₆ = 22 with 0 ≤ y₁ ≤ 7. The number of solutions with only the constraint that y₁ ≥ 0 is (22+6−16−1)=(275). To enforce the constraint that y₁ ≤ 7, we need to subtract off all the solutions in which y₁ ≥ 8.

Let y₁ = z₁ + 8. The number of solutions to Equation (2) with y₁ ≥ 8 is the same as the number of the solutions to the equation

z₁ + x₂ + x₃ + x₄ + x₅ + x₆ = 14 with z₁ ≥ 0, which is (14+6−16−1)=(195).

The number of solutions to Equation (1) with 3 ≤ x₁ ≤ 10 is (275)−(195).

Problem: solve the following one:

3 ≤ x₁ ≤ 10 and 2 ≤ x₂ ≤ 7

Answer 1

To solve the equation x₁ + x₂ + x₃ + x₄ + x₅ + x₆ = 25, with the additional constraints 3 ≤ x₁ ≤ 10 and 2 ≤ x₂ ≤ 7, we can use the given hints to simplify the equation. By substituting variables and applying the formula (n + r - 1)C(r - 1), we can find the number of solutions that satisfy the given constraints.

Step-by-step explanation:

To solve this problem, we need to find the number of solutions for the equation x₁ + x₂ + x₃ + x₄ + x₅ + x₆ = 25, where 3 ≤ x₁ ≤ 10 and 2 ≤ x₂ ≤ 7.

We can use the given hint to simplify the equation. Let's substitute x₁ with y₁ + 3. Now the equation becomes y₁ + 3 + x₂ + x₃ + x₄ + x₅ + x₆ = 25.

By converting the constraint 3 ≤ x₁ ≤ 10, we get 0 ≤ y₁ ≤ 7. Next, we can transform the equation by substituting y₁ with z₁ + 8. This gives us the equation z₁ + 8 + x₂ + x₃ + x₄ + x₅ + x₆ = 22.

To find the number of solutions, we can use the formula (n + r - 1)C(r - 1), where n is the sum of the variables and r is the number of variables. We have 22 as the sum and 6 as the number of variables, which gives us (22 + 6 - 1)C(6 - 1) = 27C5 = 27! / (5! * 22!). This gives us the number of solutions for the equation with the given constraints.