How many solutions has the equation…

Monday, 22 April 2019

6:42 PM

Machine generated alternative text: How many solutions has the equation if at2, , are non-negative integers a) with no further restriction? comb (62, 8) C8 Expected answer: 3381098545 54, 3381098545

 

Machine generated alternative text: 54 + 9 — I a) Using the dots and lines method with 54 dots and 9 lines, there are 9—1 3381098545 possible solutions.

Use the Unordered Repetition  or

 

 

 

Machine generated alternative text: b) and:rk 38 forallk= 1, 2, , 9? comb (62, 8) —comb (9, 1) *comb (23, 8) C8 Expected answer. 3376685719 3376685719

 

Substitute  where

Bring N to the RHS, then use unordered repetition, and multiply by

Subtract from (a)

Machine generated alternative text: b) Here we count the number of solutions when one variable > 38 and then apply the Inclusion/Exclusion principle. Suppose that we choose as the variable larger than 38. Then = 39 + yl whereyl 0 and original equation (39 Yl) + •••4-:r9 = 54 becomes Yl + X2 + • 15. 15 9 — Bythe dots and lines method once again, but this time with 17 dots and 9 lines, there are 9—1 9 4M)314solutions satisfying:cl > 38. ways to choose the variable which is largerthan 38, and so the number of solutions where all 38 is There are 1 54 + 9 — 1 3376685719. 9—1

 

 

Created with Microsoft OneNote 2016.