find the coefficient of x^4 in the expansion of (3x-1)^11

- anonymous

find the coefficient of x^4 in the expansion of (3x-1)^11

- chestercat

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- anonymous

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

- anonymous

and what does the 11 over 4 part mean again?

- anonymous

\[3^4=81\]
\[(-1)^7=-1\]

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## More answers

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

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

- 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

so there for it is -81x^4

- anonymous

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

- anonymous

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

- 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

thatd be great!

- 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

is my final answer -26730x^4?

- 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

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

- anonymous

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

- anonymous

can i ask u about another??!

- anonymous

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

- anonymous

sure ask away.

- anonymous

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

- 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

oops typo sorry

- anonymous

n-k=2!

- anonymous

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

- anonymous

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

- anonymous

so what would (2x)^7 be?

- anonymous

\[(2x)^7=2^7x^7\] so you will need to compute \[2^7\]

- anonymous

thats is easy, as is \[(-5)^2\]

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

so 9! over 7! times 2!

- anonymous

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

- anonymous

the answer is 36?

- anonymous

exactly!

- anonymous

so 36 times 128 times 25

- anonymous

so 25200x^7?

- 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

good!!

- anonymous

yes, 36 times 128 times 25 is it.

- anonymous

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

- anonymous

maybe i put it in wrong.

- anonymous

no i think i am right.

- anonymous

nah i put it in wrong!

- anonymous

whew i was scared but it is late.

- anonymous

is that enough of this? or are there more?

- anonymous

you have to go?

- anonymous

i can help you with another if you like.

- anonymous

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

- anonymous

final answer yes.

- anonymous

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

- 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

awesome thanks!! haha

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