GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
306 views

Find the coefficient of x83 in (x5+ x8+ x11+ x14+ x17)10 ?

asked in Combinatory by Veteran (11.5k points)   | 306 views
can somebody provide me exact answer........

Debashish Deka's ans is correct! What else you need?

2 Answers

+11 votes
Best answer

$\begin{align*} &\left [ {\color{red}{\bf x^{83}}} \right ] : \left [ \left ( x^5+x^8 + x^{11} + x^{14} + x^{17}\right )^{10} \right ] \\ &\left [ {\color{red}{\bf x^{83}}} \right ] :\left [ x^{50}\left ( 1+x^3 + x^{6} + x^{9} + x^{12} \right )^{10} \right ] \\ &\left [ {\color{red}{\bf x^{33}}} \right ] :\left [\left ( 1+x^3 + x^{6} + x^{9} + x^{12} \right )^{10} \right ] \\ &\left [ {\color{red}{\bf x^{33}}} \right ] :\left [ \left \{ 1+(x^3)^1 + (x^{3})^2 + (x^{3})^3 + (x^{3})^4 \right \}^{10} \right ] \\ &\left [ {\color{red}{\bf x^{33}}} \right ] : \left [ \left \{ \frac{1-(x^3)^{4+1}}{1-(x^3)} \right \}^{10} \right ] \\ &\left [ {\color{red}{\bf x^{33}}} \right ] : \left [ \left ( 1-x^{15} \right )^{10}.\frac{1}{\left ( 1-x^3 \right )^{10}} \right ] \\ &\left [ {\color{red}{\bf x^{33}}} \right ] : \left [ \sum_{r=0}^{10}\binom{10}{r}\left ( -x^{15} \right )^r \; . \; \sum_{k=0}^{\infty}\binom{10+k-1}{k} . (x^3)^k \right ] \\ \end{align*}$

 

Now,

$\begin{align*} &\left [ {\color{red}{\bf x^{33}}} \right ] : \left [ \sum_{r=0}^{10}\binom{10}{r}\left ( -x^{15} \right )^r \; . \; \sum_{k=0}^{\infty}\binom{10+k-1}{k} . (x^3)^k \right ] \\ \\ &\left [ {\color{red}{\bf x^{33}}} \right ] : \begin{cases} (-1)^0.\binom{10}{0}.\binom{10+11-1}{11} \qquad r = 0, k = 11 \\ \\ (-1)^1.\binom{10}{1}.\binom{10+6-1}{6} \qquad r = 1, k = 6 \\ \\ (-1)^2.\binom{10}{2}.\binom{10+1-1}{1} \qquad r = 2, k = 1 \\ \end{cases} \\ \\ &\left [ {\color{red}{\bf x^{33}}} \right ] : \begin{cases} +\binom{10}{0}.\binom{20}{11} \qquad r = 0, k = 11 \\ \\ -\binom{10}{1}.\binom{15}{6} \qquad r = 1, k = 6 \\ \\ +\binom{10}{2}.\binom{10}{1} \qquad r = 2, k = 1 \\ \end{cases} \end{align*}$

 


 

NOTE:

1. $1+x+x^2+x^3+.....x^n = \frac{1-x^{n+1}}{1-x}$

2. $\frac{1}{(1-x)^n} = \sum_{r=0}^{\infty}\binom{n+r-1}{r}.x^r$

3. $\left [ x^{83} \right ]$ means coefficient of $x^{83}$ of the whole expression.

answered by Veteran (47.1k points)  
selected by
+4 votes

Coefficient of x83 in (x5+ x8+ x11+ x14+ x17)10 

= Coefficient of x33 in (1 + x+ x+ x+ x12)   [Just took x5 common so it's x50(1 + x+ x+ x+ x12)10 ]

= Coefficient of x33 in [1-(x3)5 / 1-x ]^10 [Applied sum of GP i.e a(1-rn)/(1-r) where r is common ration and a is first term]

=  Coefficient of x33 in (1 - x15 )10 (1 - x)-10 

= 10C0 * (-1)0 * -10C11 * (-1)11 10C1 * (-1)1 * -10C6 * (-1)610C2 * (-1)2 * -10C1 * (-1)--> now it can be solved :)

To solve futher use extended bionomial theorem ie -nCr  =  (n+r-1)C* (-1)r

answered by Loyal (3.5k points)  
is the answer = 1294080

20C11  - 10* 15C6  + 45* 10C1


Top Users May 2017
  1. akash.dinkar12

    3548 Points

  2. pawan kumarln

    2126 Points

  3. Bikram

    1922 Points

  4. sh!va

    1682 Points

  5. Arjun

    1622 Points

  6. Devshree Dubey

    1272 Points

  7. Debashish Deka

    1270 Points

  8. Angkit

    1056 Points

  9. LeenSharma

    1028 Points

  10. Arnab Bhadra

    812 Points

Monthly Topper: Rs. 500 gift card
Top Users 2017 May 22 - 28
  1. Bikram

    1008 Points

  2. pawan kumarln

    752 Points

  3. Arnab Bhadra

    726 Points

  4. akash.dinkar12

    428 Points

  5. Arjun

    350 Points


22,896 questions
29,206 answers
65,329 comments
27,708 users