anonymous
  • anonymous
find the coefficient of x^4 in the expansion of (3x-1)^11
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
same as before \[\dbinom{11}{4}(3x)^4(-1)^7\]
anonymous
  • anonymous
and what does the 11 over 4 part mean again?
anonymous
  • anonymous
\[3^4=81\] \[(-1)^7=-1\]

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anonymous
  • anonymous
do you know how to find \[\dbinom{11}{4}\]? if not i will be happy to show you.
anonymous
  • anonymous
it is called "eleven choose 4" the number of ways to choose 4 items out of 11.
anonymous
  • anonymous
show please!! also why did u just do 3^4 isnt ther an x in there like 3x^4?
anonymous
  • anonymous
\[(3x)^4=3^4x^4\] and since you were asked for the coefficient that is the number you need to compute.
anonymous
  • anonymous
it is the whole thing raised to the power of 4, not just the x. that is it is \[(3x)^4\] not \[3x^4\]
anonymous
  • anonymous
so there for it is -81x^4
anonymous
  • anonymous
if i times it by the -1 like i was supposed to?
anonymous
  • anonymous
now to compute \[\dbinom{11}{4}\] yes there is a -81
anonymous
  • 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
anonymous
  • anonymous
thatd be great!
anonymous
  • 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 \[\dbinom{11}{4}=\frac{11\times 10 \times 9\times 8}{4 \times 3\times 2}\]
anonymous
  • anonymous
is my final answer -26730x^4?
anonymous
  • anonymous
now since this is a whole number , cancel first and multiply last! \[\frac{11 \times 10\times 9 \times 8}{4\times 3\times 2}={11\times 10\times 3}\]
anonymous
  • anonymous
?
anonymous
  • anonymous
\[\dbinom{11}{4}=330\] \[330\times 81 \times -1=-26730\] yes you got it!
anonymous
  • anonymous
thankyou so much!!!!!!! your awesome!
anonymous
  • anonymous
can i ask u about another??!
anonymous
  • anonymous
no problem. try \[\dbinom{10}{3}\] and convince yourself that it is the same as \[\dbinom{10}{7}\]because 3+7=10!
anonymous
  • anonymous
sure ask away.
anonymous
  • anonymous
find the coefficient of x^7 in the expansion of (2x-5)^9
anonymous
  • anonymous
ok same idea. here n = 9, k = 7, n-k=3 so the term with \[x^7\] will look like \[\dbinom{9}{7}(2x)^7(-5)^3\]
anonymous
  • anonymous
oops typo sorry
anonymous
  • anonymous
n-k=2!
anonymous
  • anonymous
my mistake. it will be \[\dbinom{9}{7}(2x)^7(-5)^2\]
anonymous
  • anonymous
exponents have to add up to 9. would you like to try it?
anonymous
  • anonymous
so what would (2x)^7 be?
anonymous
  • anonymous
\[(2x)^7=2^7x^7\] so you will need to compute \[2^7\]
anonymous
  • anonymous
thats is easy, as is \[(-5)^2\]
anonymous
  • anonymous
your real job is to compute \[\dbinom{9}{7}\]
anonymous
  • anonymous
yes how do u do the 9 over 7 thging again?
anonymous
  • anonymous
\[\dbinom{n}{k}=\frac{n!}{k! (n-k)!}\]
anonymous
  • anonymous
here n = 9, k = 7 and n-k=2
anonymous
  • anonymous
so 9! over 7! times 2!
anonymous
  • anonymous
right. but don't forget to cancel away first because the entire denominator will cancel
anonymous
  • anonymous
the answer is 36?
anonymous
  • anonymous
exactly!
anonymous
  • anonymous
so 36 times 128 times 25
anonymous
  • anonymous
so 25200x^7?
anonymous
  • 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. \[\dbinom{9}{2}=\frac{9\times 8}{2}=9\times 4=36\]
anonymous
  • anonymous
good!!
anonymous
  • anonymous
yes, 36 times 128 times 25 is it.
anonymous
  • anonymous
oh wait i got a different number than you. i got 115200
anonymous
  • anonymous
maybe i put it in wrong.
anonymous
  • anonymous
no i think i am right.
anonymous
  • anonymous
nah i put it in wrong!
anonymous
  • anonymous
whew i was scared but it is late.
anonymous
  • anonymous
is that enough of this? or are there more?
anonymous
  • anonymous
you have to go?
anonymous
  • anonymous
i can help you with another if you like.
anonymous
  • anonymous
one question quick.. the fianl answer is 115200x^7 rights?
anonymous
  • anonymous
final answer yes.
anonymous
  • anonymous
nah its okay i will let u go! u were a great help! quick question .. ur suyper smart!! how old are u?!
anonymous
  • 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
anonymous
  • anonymous
awesome thanks!! haha

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