Final answer:
a. 14,190 solutions, b. 14,190 solutions, c. 178,365 solutions, d. 14,190 solutions, e. 13,270 solutions.
Step-by-step explanation:
a.
For this equation, we can use the stars and bars method to count the number of solutions. Since all xi values are greater than 0, we can subtract 1 from each xi to make them positive integers. Then, we have the equation (x1-1)+(x2-1)+(x3-1)+(x4-1)+(x5-1)=58. This is a problem of distributing 58 identical objects into 5 distinct boxes, which has a total of 57+5-1C5-1 = 61C4 = 14,190 solutions.
b.
For this equation, we have the same number of solutions as the previous case, as x₁ can be 0.
c.
For this inequality, we can use the stars and bars method to count the number of solutions. The only difference is that we can have one of the xi values equal to 0. This means we need to find the number of non-negative integer solutions to the equation x1+x2+x3+x4+x5=63, which is 63+5-1C5-1 = 67C4 = 178,365 solutions.
d.
Since all xi values must be greater than 0, we can subtract 1 from each xi to make them positive integers. Then, we have the equation (x1-1)+(x2-1)+(x3-1)+(x4-1)+(x5-1)=58, which has 61C4 = 14,190 solutions.
e.
Since all xi values must be greater than 0 and x2 must be less than or equal to 9, we can subtract 1 from each xi to make them positive integers. Then, we have the equation (x1-1)+(x2-1)+(x3-1)+(x4-1)+(x5-1)=58, where x2-1 must be less than or equal to 8. We can count the number of solutions using the stars and bars method, which gives us 61+4C4+58+3C3+58+5-1C5-1 = 13,270 solutions.