How many integer solutions of x1 + x2 + … x6 = 20 satisfies 1 ≤ xi ≤ 9? (use inclusion exclusion principle)
To find the number of integer solutions that satisfy the given conditions using the inclusion-exclusion principle, we need to consider all possible combinations and then subtract the combinations that violate the conditions.
Let's break down the problem into smaller steps.
Step 1: Find the number of integer solutions without any constraints.
We can think of this problem as distributing 20 identical balls to 6 distinct boxes. We can use a stars and bars approach to find the number of solutions. Since there are 5 "gaps" between the balls and boxes, we need to place 5 dividers among them. Using the stars and bars formula, the number of solutions is:
(20 + 5 - 1) C (5) = 24 C 5.
Step 2: Find the number of integer solutions where some xi is greater than 9.
To identify the number of solutions that violate the condition, we can consider each xi individually. If xi > 9, we subtract 10 from xi and treat it as a new variable yi, where 1 ≤ yi ≤ 9. Then, we solve the equation:
y1 + y2 + ... y6 = (20 - 10k)
where k represents the number of variables that exceed 9.
The number of solutions can be found using the stars and bars formula as:
((20 - 10k) + 5 - 1) C (5).
Step 3: Apply the inclusion-exclusion principle.
To find the number of solutions that satisfy the conditions, we subtract the number of solutions where at least one variable exceeds 9.
Number of Solutions = Total Solutions - Solutions violating the condition
Number of Solutions = 24 C 5 - ((20 - 10*1) + 5 - 1) C (5) + ((20 - 10*2) + 5 - 1) C (5) - ((20 - 10*3) + 5 - 1) C (5) + ...
Simplifying further will give you the final answer.