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- anonymous

what is (2x+3) times itself 10 times?

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- anonymous

what is (2x+3) times itself 10 times?

- chestercat

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- anonymous

x= 5

- anonymous

you mean \[(2x+3)^{10}\]?

- anonymous

perhaps this is the answer or perhaps you are being asked to expand?

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- anonymous

alright well the original question is " find the coefficient of x^6 in the expansion of (2x+3)^10

- anonymous

ahh that is a different story

- anonymous

because writing out the entire expansion is a pain.

- anonymous

Remember this formula for co-efficient during expansion:
1 1 = Power 1
1 2 1 = Power 2 (Retain the 1st number, add 1st and 2nd number put as 2nd, retain 2nd number as 3rd)
1 3 3 1 = Power 3 (Retain the 1st number, add 1st and 2nd number put as 2nd, add 2nd and 3rd as 3rd, retain 3rd number as 4th)
1 4 6 4 1 = Power 4
1 5 10 10 5 1 = Power 5
Hence the coefficient of x³ is: 10*(-2)³ *3²= -80 * 9 = -720
Note: Interestingly:
(1) The terms for power 1 is 2, for power2 it is 3, for power n the terms are n + 2
(2) The sum of the coefficients for power 1 is: 1+1 = 2 = 2¹, for power 2, 1+2+1 = 4 = 2², power 3 is 1+3+3+1=2³,..., for power n, it is 2^n

- anonymous

well thats what i have to do is find the coefficient of x^6 in the expansion of (2x+3)^10

- anonymous

ohh kk

- anonymous

\[\dbinom{10}{6}(2x)^6\]

- anonymous

^^ idk what that means....

- anonymous

The coefficient of x^5 is -8064
The x^6 has a coefficient of -13608

- anonymous

\[\dbinom{10}{6}=\frac{10\times 9 \times 8\times 7}{4\times 3\times 2}=210\]

- anonymous

\[2^6=64\]

- anonymous

wait wat
that like wrong

- anonymous

maybe . i am tired. why?
i thought it was 64*210=13440

- anonymous

oh carp. i am wrong. i forgot \[3^4\]
that is there too!

- anonymous

im sorry i have no idea what the H ur talking about

- anonymous

term with \[x^6\]
looks like \[\dbinom{10}{6}(2x^6)(3^4)\]

- anonymous

if you expand \[(a+b)^n\] the terms look like \[\dbinom{n}{k}a^kb^{n-k}\]
here n= 10
k=6
a = 2x
b=3

- anonymous

ohhh okay haha thanks

- anonymous

so you get \[\dbinom{10}{6}(2x)^63^4\]

- anonymous

\[\dbinom{10}{6}=210\]
\[2^6=64\]
\[3^4=81\]
you have to multiply all this mess together. sorry i screwed up at the beginning.

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