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

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

asked in Combinatory by Veteran (10.4k points)   | 138 views
can somebody provide me exact answer........

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

2 Answers

+8 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 (36.5k points)  
selected by
+3 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 (3.3k points)  
Top Users Jan 2017
  1. Debashish Deka

    8630 Points

  2. sudsho

    5406 Points

  3. Habibkhan

    4798 Points

  4. Bikram

    4532 Points

  5. Vijay Thakur

    4482 Points

  6. saurabh rai

    4222 Points

  7. Arjun

    4156 Points

  8. santhoshdevulapally

    3760 Points

  9. Sushant Gokhale

    3576 Points

  10. GateSet

    3398 Points

Monthly Topper: Rs. 500 gift card

19,195 questions
24,079 answers
53,024 comments
20,314 users