Consider the non negative integer solutions for the equation $$ x_1 +
x_2 + x_3 + x_4 + x_5 = 36. $$(a) How many distinct solutions are there?
(b) How many distinct solutions are there if $x_1 \ge 12$ ?
(c) How many distinct solutions are there if $x_1 < 18$ ?
(d) How many distinct solutions are there if $x_1 < 18$ and $x_2 < 6$ ?
My working:
a) $\frac{5!}{1!} = 150 $
Not sure how to approach this question.
Best Answer
The first part is a classical Stars and Bars problem.
For the second part substitute $x_6 = x_1 + 12$ and apply the Stars and Bars reasoning for $x_2 + x_3 + x_4 + x_5 + x_6 = 24$
For the third part do same as for part two, but this time substitute $x_6 = x_1 + 18$ and then subtract the number you will get from the number of total solutions, which we obtained in part one.
The final part is similar to the part three, but now you will need a little help from the Inclusion-Exclusion Principle