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# anyone, help!

+1
365
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1. Find the coefficient of $$x^3y^3z^2$$ in the expansion of $$(x+y+z)^8$$.

2. For each integer $$n$$, let $$f(n)$$ be the sum of the elements of the $$n$$ th row (i.e. the row with $$n+1$$ elements) of Pascal's triangle minus the sum of all the elements from previous rows. For example, $$f(2) = \underbrace{(1 + 2 + 1)}_{\text{2nd row}} - \underbrace{(1 + 1 + 1)}_{\text{0th and 1st rows}} = 1.$$ What is the minimum value of $$f(n)$$ for $$n \ge 2015$$?

Mar 8, 2018

#1
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Here's the first, tertre

( x + y + z)^8  =   ( x  +  ( y + z) )^8  =

C (8,5) * x^3 * (y + z)^5   .......we need to  find the coefficiient on y^3z^2  for the second part in red

C (8,5) * x^3 * C(5,3)* y^3 * z^2  =

56* x^3  *  10 * y^3 * z^2   =

560   (x^3 * y^3 * z^2)

Mar 8, 2018
edited by CPhill  Mar 8, 2018
#2
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Just a minor typo in the 3rd to the last step. The second x^3 should read y^3.

Mar 8, 2018
#3
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It will always be $$2^n-(2^{n-1}+2^{n-2} \cdots +2^1+2^0)=1$$.Thus, the minimum value is $$\boxed{1}$$

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Mar 8, 2018