2 votes 2 votes How many solutions are there to the equation x1 + x2 + x3 + x4 + x5 = 21, where xi , i = 1, 2, 3, 4, 5, is a nonnegative integer such that: 0 ≤ x1 ≤ 3 , 1 ≤ x2 < 4 and 15≤x3 ? Combinatory generating-functions discrete-mathematics + – rahul sharma 5 asked Jun 10, 2017 rahul sharma 5 2.6k views answer comment Share Follow See all 4 Comments See all 4 4 Comments reply rahul sharma 5 commented Jun 10, 2017 reply Follow Share I tried moving lower constraint from x2(1) and x3(15) and RHS becomes 21-6=5. Then i find the coefficient of x^5 which is coming as -106 but answer in book is 106,Does sign matters here?I know number of solutions will be +ve only but if it comes -ve then can we ignore sign or do we give 0 as answer? 0 votes 0 votes Hemant Parihar commented Jun 10, 2017 reply Follow Share $\binom{24}{4}$ - $\binom{8}{4}$ - $\binom{21}{4}$ + $\binom{5}{4}$ - $\binom{20}{4}$ + $\binom{4}{4}$ + $\binom{17}{4}$ ???? 0 votes 0 votes rahul sharma 5 commented Jun 10, 2017 reply Follow Share Given answer is 106. 0 votes 0 votes Hemant Parihar commented Jun 11, 2017 reply Follow Share $\binom{9}{4}$ - $\binom{6}{4}$ - $\binom{5}{4}$ = 106. Sorry by mistake in the previous one I considered x3 <= 15. 0 votes 0 votes Please log in or register to add a comment.
4 votes 4 votes Hope this helps. Pinaki Dash answered Jun 10, 2017 Pinaki Dash comment Share Follow See all 3 Comments See all 3 3 Comments reply rahul sharma 5 commented Jun 10, 2017 reply Follow Share I started with the same approach. I have written all the terms in power of x.And then i find coefficient of x^5 .And it came as -106.So we can ignore the coefficient and select 106 as the answer? If you didn't understand my approach,i will upload an image.let me know 0 votes 0 votes iarnav commented Jul 21, 2017 reply Follow Share How y2 = x2-1? 0 votes 0 votes Pinaki Dash commented Jul 24, 2017 reply Follow Share Given that x2>=1. So by adding -1 on both sides of inequality: (x2-1)>=(1-1) => (x2-1)>=0. Now instead of writing x2-1 everywhere further, replace it with y2 such that y2=x2-1 and y2>=0. 1 votes 1 votes Please log in or register to add a comment.
2 votes 2 votes answer: -106 Prateek kumar answered Jan 28, 2018 Prateek kumar comment Share Follow See all 0 reply Please log in or register to add a comment.