How many integral solutions are there to x1 + x2 + x3 + x4 + x5 + x6 = 20, where x1 ≥ 3, x2 ≥ 2, x3 ≥ 2, x4 ≥ 2, x5 ≥ 1, x6 ≥ 0?

Steps to Solve

1. Define the variables with minimum values

We have the equation ( z_1 + z_2 + z_3 + z_4 + z_5 + z_6 = 20 ) with constraints:

• ( z_1 \geq 3 )
• ( z_2 \geq 2 )
• ( z_3 \geq 2 )
• ( z_4 \geq 4 )
• ( z_5 \geq 2 )
• ( z_6 \geq 0 )

2. Adjust for the minimum values

To simplify the problem, we will redefine the variables:

• Let ( z_1' = z_1 - 3 ) (hence ( z_1' \geq 0 ))
• Let ( z_2' = z_2 - 2 ) (hence ( z_2' \geq 0 ))
• Let ( z_3' = z_3 - 2 ) (hence ( z_3' \geq 0 ))
• Let ( z_4' = z_4 - 4 ) (hence ( z_4' \geq 0 ))
• Let ( z_5' = z_5 - 2 ) (hence ( z_5' \geq 0 ))
• ( z_6' = z_6 ) (hence ( z_6' \geq 0 ))

Now, we can substitute these back into the equation:

$$ z_1' + z_2' + z_3' + z_4' + z_5' + z_6' + 3 + 2 + 2 + 4 + 2 + 0 = 20 $$

Thus, we simplify it to:

$$ z_1' + z_2' + z_3' + z_4' + z_5' + z_6' = 20 - (3 + 2 + 2 + 4 + 2) $$

3. Calculate the new equation

Calculating the right-hand side gives:

$$ 20 - 13 = 7 $$

So we have:

$$ z_1' + z_2' + z_3' + z_4' + z_5' + z_6' = 7 $$

4. Apply the stars and bars method

The number of non-negative integer solutions to the equation

$$ x_1 + x_2 + ... + x_k = n $$

is given by the formula:

$$ \binom{n + k - 1}{k - 1} $$

where ( n ) is the total and ( k ) is the number of variables.

In our case, ( n = 7 ) and ( k = 6 ):

5. Substitute into the formula

Now, we can find the number of solutions:

$$ \binom{7 + 6 - 1}{6 - 1} = \binom{12}{5} $$

6. Calculate the binomial coefficient

Finally, calculate ( \binom{12}{5} ):

$$ \binom{12}{5} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} = 792 $$