Find the number of solutions in integers to n 1 + n 2 + n...

woowheedr

woowheedr

Answered

2022-07-08

Find the number of solutions in integers to n 1 + n 2 + n 3 + n 4 = 12 satisfying 0 n 1 4 , 0 n 2 5 , 0 n 3 8 , and  0 n 4 9.

Answer & Explanation

grubijanebb

grubijanebb

Expert

2022-07-09Added 10 answers

Step 1
Let A denote nonnegative solutions to n 1 + n 2 + n 3 + n 4 = 12 and A i denote those solutions where the upper bound on n i is violated, then inclusion/exclusion gives | A 1 A 2 A 3 A 4 ¯ | = | A | ( | A 1 | + | A 2 | + | A 3 | + | A 4 | ) + ( | A 1 A 2 | + ) = ( 12 + 4 1 12 ) ( ( 7 + 4 1 7 ) + ( 6 + 4 1 6 ) + ( 3 + 4 1 3 ) + ( 2 + 4 1 2 ) ) + ( 1 + 4 1 1 ) where e.g. | A 1 | = ( 7 + 4 1 7 )
Step 2
Because the solutions there can be recast as solutions to ( m 1 + 5 ) + n 2 + n 3 + n 4 = 12 with m 1 , n i 0 or m 1 + n 2 + n 3 + n 4 = 7, and omitted terms are because they are empty (i.e. they reduce to a sum of nonnegative integers to a negative integer).

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?