# 1.) Use the binomial theorem to write the binomial expansion of `(1/2x+3y)^4` 2.) Use the binomial theorem to find the 18th term in the binomial expansion of `(2x-ysqrt(2))^27` ` ` 3.) Find the...

1.) Use the binomial theorem to write the binomial expansion of `(1/2x+3y)^4`

2.) Use the binomial theorem to find the 18th term in the binomial expansion of `(2x-ysqrt(2))^27`

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3.) Find the 69th number in the 72nd row (n=72) of Pascal's triangle.

4.) Refer to the photo

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### 1 Answer

embizze | High School Teacher | (Level 2) Educator Emeritus

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(1) Expand `(1/2 x +3y)^4 ` :

`= ([4],[0])(1/2x)^4(3y)^0+([4],[1])(1/2x)^3(3y)^1+([4],[2])(1/2x)^2(3y)^2+([4],[3])(1/2x)^1(3y)^3 + ([4],[4])(1/2x)^0(3y)^4 ` `=(1)(1/16 x^4)(1)+4(1/8 x^3)(3y)+(6)(1/4 x^2)(9y^2)+4(1/2 x)(27y^3)+(1)(1)(81y^4) ` `=1/16x^4+3/2x^3y+27/2x^2y^2+54xy^3+81y^4 `

Note that `([4],[2])= ` `_4 C _2 ` or the combinations of 4 objects taken 2 at a time.

(2) Find the 18th term of `(2x-ysqrt(2))^27 ` :

`([27],[17])(2x)^10(-ysqrt(2))^17 `

or `(8436285)(1024x^10)(-256sqrt(2)y^17)~~-2.21"x"10^12x^(10)y^(17) `

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