How do you expand ${(d - 5)}^{6}$ $(d - 5)^6$ using Pascal’s Triangle?

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How do you expand ${(d - 5)}^{6}$ $(d - 5)^6$ using Pascal’s Triangle?

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Answer 1 · 2015-10-02T17:54:24

See the explanation below.

Explanation:

Use the 7th row:
$1615201561$ $1" "6" "15" "20" "15" "6" "1$ to get the coefficients.

Powers on $d$ $d$ are decreasing from $6$ $6$ to $0$ $0$ and powers on $- 5$ $-5$ are increasing from $0$ $0$ to $6$ $6$ .

${(d - 5)}^{6} = 1 d^{6} + 6 d^{5} {(- 5)}^{1} + 15 d^{4} {(- 5)}^{2} + 20 d^{3} {(- 5)}^{3} + 15 d^{2} {(- 5)}^{4} + 6 d^{1} {(- 5)}^{5} + 1 {(- 5)}^{6}$ $(d-5)^6 = 1d^6+6d^5(-5)^1+15d^4(-5)^2+20d^3(-5)^3+15d^2(-5)^4+6d^1(-5)^5+1(-5)^6$

$= d^{6} + 6 d^{5} (- 5) + 15 d^{4} (25) + 20 d^{3} (- 125) + 15 d^{2} (625) + 6 d^{1} (- 3125) + 1 (15625)$ $=d^6+6d^5(-5)+15d^4(25)+20d^3(-125)+15d^2(625)+6d^1(-3125)+1(15 625)$

$= d^{6} - 30 d^{5} + 375 d^{4} - 2500 d^{3} + 9375 d^{2} - 18750 d + 15625$ $= d^6-30d^5+375d^4-2500d^3+9375d^2-18750d+15 625$

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How do you expand ${(d - 5)}^{6}$ $(d - 5)^6$ using Pascal’s Triangle?

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