SOLUTION: Find the coefficient of x^6 in the expansion of (2x-3)^9

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Question 1181279: Find the coefficient of x^6 in the expansion of (2x-3)^9
Answer by ikleyn(52795) About Me  (Show Source):
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Find the coefficient of x^6 in the expansion of (2x-3)^9
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The binomial expansion is this formula


    %28a%2Bb%29%5En = a%5En + C%5Bn%5D%5E1%2Aa%5E%28n-1%29%2Ab + C%5Bn%5D%5E2%2Aa%5E%28n-2%29%2Ab%5E2 + C%5Bn%5D%5E3%2Aa%5E%28n-3%29%2Ab%5E3 + . . . + C%5Bn%5D%5E%28n-1%29%2Aa%5E1%2Ab%5E%28n-1%29 + b%5En


In our case,  n = 9,  a = 2x,  b = -3.  Therefore, the general term of the binomial expansion in our case is  


    C%5B9%5D%5Ek%2A%282x%29%5E%289-k%29%2A%28-3%29%5Ek, k = 0, 1, 2, 3, 4, 5, . . . 


Then the term with  x^6  is the fourth term of the expansion at k = 3 


    C%5B9%5D%5E3%2A%282x%29%5E%289-3%29%2A%28-3%29%5E3 = %28%289%2A8%2A7%29%2F%281%2A2%2A3%29%29%2A2%5E6%2Ax%5E6%2A%28-3%29%5E3 = -145162%2Ax%5E6.    ANSWER

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If you want to see other similar solved problems and to learn the subject wider and deeper,  then look into the lesson
    - Solved problems on binomial coefficients
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this textbook under the topic
"Binomial expansion, binomial coefficients, Pascal's triangle".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.