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To expand the binomial (d–5y)6 using Pascal's triangle, we need to find the coefficients of the terms in the expansion. Pascal's triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it.
The 6th row of Pascal's triangle is 1, 6, 15, 20, 15, 6, 1
To expand (d–5y)6, we can write it as:
1(d)6 + 6(d)5(-5y) + 15(d)4(-5y)2 + 20(d)3(-5y)3 + 15(d)2(-5y)4 + 6(d)(-5y)5 + 1(-5y)6
Simplifying each term, we get:
(d)6 + 6(d)5(-5y) + 15(d)4(25y2) + 20(d)3(-125y3) + 15(d)2(625y4) + 6(d)(-3125y5) + (-625y)6
This can be further simplified as:
d^6 - 30d^5y + 750d^4y^2 - 10,000d^3y^3 + 93,750d^2y^4 - 187,500dy^5 + 156,250y^6
Therefore, the expansion of (d–5y)6 using Pascal's triangle is:
d^6 - 30d^5y + 750d^4y^2 - 10,000d^3y^3 + 93,750d^2y^4 - 187,500dy^5 + 156,250y^6