## anonymous 5 years ago find the coefficient of x^4 in the expansion of (3x-1)^11

1. anonymous

same as before $\dbinom{11}{4}(3x)^4(-1)^7$

2. anonymous

and what does the 11 over 4 part mean again?

3. anonymous

$3^4=81$ $(-1)^7=-1$

4. anonymous

do you know how to find $\dbinom{11}{4}$? if not i will be happy to show you.

5. anonymous

it is called "eleven choose 4" the number of ways to choose 4 items out of 11.

6. anonymous

show please!! also why did u just do 3^4 isnt ther an x in there like 3x^4?

7. anonymous

$(3x)^4=3^4x^4$ and since you were asked for the coefficient that is the number you need to compute.

8. anonymous

it is the whole thing raised to the power of 4, not just the x. that is it is $(3x)^4$ not $3x^4$

9. anonymous

so there for it is -81x^4

10. anonymous

if i times it by the -1 like i was supposed to?

11. anonymous

now to compute $\dbinom{11}{4}$ yes there is a -81

12. anonymous

but you have to multiply by $\dbinom{11}{4}$ which is easy enough to compute. i can show you step by step if you like

13. anonymous

thatd be great!

14. anonymous

ok first of all a formula, although you don't really use it. the formula is $\dbinom{n}{k}=\frac{n!}{k! (n-k)!}$ here n = 11, k = 4 and n-k=7 make a fraction. in the numerator put 4 numbers starting at 11 and counting down. in the denominator you put 4! to get $\dbinom{11}{4}=\frac{11\times 10 \times 9\times 8}{4 \times 3\times 2}$

15. anonymous

is my final answer -26730x^4?

16. anonymous

now since this is a whole number , cancel first and multiply last! $\frac{11 \times 10\times 9 \times 8}{4\times 3\times 2}={11\times 10\times 3}$

17. anonymous

?

18. anonymous

$\dbinom{11}{4}=330$ $330\times 81 \times -1=-26730$ yes you got it!

19. anonymous

thankyou so much!!!!!!! your awesome!

20. anonymous

21. anonymous

no problem. try $\dbinom{10}{3}$ and convince yourself that it is the same as $\dbinom{10}{7}$because 3+7=10!

22. anonymous

23. anonymous

find the coefficient of x^7 in the expansion of (2x-5)^9

24. anonymous

ok same idea. here n = 9, k = 7, n-k=3 so the term with $x^7$ will look like $\dbinom{9}{7}(2x)^7(-5)^3$

25. anonymous

oops typo sorry

26. anonymous

n-k=2!

27. anonymous

my mistake. it will be $\dbinom{9}{7}(2x)^7(-5)^2$

28. anonymous

exponents have to add up to 9. would you like to try it?

29. anonymous

so what would (2x)^7 be?

30. anonymous

$(2x)^7=2^7x^7$ so you will need to compute $2^7$

31. anonymous

thats is easy, as is $(-5)^2$

32. anonymous

your real job is to compute $\dbinom{9}{7}$

33. anonymous

yes how do u do the 9 over 7 thging again?

34. anonymous

$\dbinom{n}{k}=\frac{n!}{k! (n-k)!}$

35. anonymous

here n = 9, k = 7 and n-k=2

36. anonymous

so 9! over 7! times 2!

37. anonymous

right. but don't forget to cancel away first because the entire denominator will cancel

38. anonymous

the answer is 36?

39. anonymous

exactly!

40. anonymous

so 36 times 128 times 25

41. anonymous

so 25200x^7?

42. anonymous

now i show you the easy way. first of all 7+2=9 so it is easier to compute $\dbinom{9}{2}$ so we work as before. make a fraction. in the numerator put to numbers starting at 9 in the denominator put 2. we get the answer right away. $\dbinom{9}{2}=\frac{9\times 8}{2}=9\times 4=36$

43. anonymous

good!!

44. anonymous

yes, 36 times 128 times 25 is it.

45. anonymous

oh wait i got a different number than you. i got 115200

46. anonymous

maybe i put it in wrong.

47. anonymous

no i think i am right.

48. anonymous

nah i put it in wrong!

49. anonymous

whew i was scared but it is late.

50. anonymous

is that enough of this? or are there more?

51. anonymous

you have to go?

52. anonymous

i can help you with another if you like.

53. anonymous

one question quick.. the fianl answer is 115200x^7 rights?

54. anonymous

55. anonymous

nah its okay i will let u go! u were a great help! quick question .. ur suyper smart!! how old are u?!

56. anonymous

old as black pepper. have fun, and don't forget to convince yourself that $\dbinom{10}{6}=\dbinom{10}{4}$because 10=4+6

57. anonymous

awesome thanks!! haha