

InterviewSolution
Saved Bookmarks
1. |
The value of .^(n)C_(0) xx .^(2n)C_(r) - .^(n)C_(1)xx.^(2n-2)C_(r)+.^(n)C_(2)xx.^(2n-4)C_(r)+"…." is equal to |
Answer» <html><body><p>`.^(n)C_(r-n) <a href="https://interviewquestions.tuteehub.com/tag/xx-747671" style="font-weight:bold;" target="_blank" title="Click to know more about XX">XX</a> 2^(2n-r)` if `r ge n`<br/><a href="https://interviewquestions.tuteehub.com/tag/0-251616" style="font-weight:bold;" target="_blank" title="Click to know more about 0">0</a>, if `r lt n`<br/>`.^(n)C_(r-n) xx 2^(n-r)` if `r ge n`<br/>`.^(-n)C_(r-n) xx 2^(2n-r)` if `r lt n`</p><a href="https://interviewquestions.tuteehub.com/tag/solution-25781" style="font-weight:bold;" target="_blank" title="Click to know more about SOLUTION">SOLUTION</a> :`.^(n)C_(0)xx.^(2n-r)C_(r)-.^(n)C_(1) xx .^(2n-2)C_(r)+.^(n)C_(2)xx.^(2n-4)C_(r)+"...."` <br/> = Coefficient of `<a href="https://interviewquestions.tuteehub.com/tag/x-746616" style="font-weight:bold;" target="_blank" title="Click to know more about X">X</a>^(r)` in `(.^(n)C_(0) (1+x)^(2n)-.^(n)C_(1) (1+x)^(2n-2) + .^(n)C_(2)(1+x)^(2n-4)+"…..")` <br/> = Coefficient of `x^(r)` in `(.^(n)C_(0)((1+x)^(2))^(n) - .^(n)C_(1)((1+x)^(2))^(n-1)+.^(n)C_(2)((1+x)^(2))^(n-2)+"....")` <br/> `=` Coefficient of `x^(r)` in `((1+x)^(2)-1)^(n)` <br/> = Coefficient of `x^(r)` in `(<a href="https://interviewquestions.tuteehub.com/tag/2x-301182" style="font-weight:bold;" target="_blank" title="Click to know more about 2X">2X</a>+x^(2))^(n)` <br/> General term, `T_(p+1) = .^(n)C_(p)(2x)^(n-p)(x^(2))^(p) = .^(n)C_(p)2^(n-p)x^(n+p)` <br/> For `x^(r ), n+p=r`, So, `p = r-n`. <br/> `:.` Coefficient of `x^(r) = .^(n)C_(r-n) xx 2^(2n-r)` <br/> This is non-zero if `r ge n` and zero if `r lt n`.</body></html> | |