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Use Pascal’s triangle to expand the binomial.
(d–5y)^6
To expand the binomial (d-5y)^6 using Pascal's triangle, we will use the formula for expanding a binomial with an exponent:
(a + b)^n = C(n, 0)*a^(n-0)*b^0 + C(n, 1)*a^(n-1)*b^1 + C(n, 2)*a^(n-2)*b^2 + ... + C(n, n)*a^(n-n)*b^n
where C(n, r) represents the combination of choosing r items from a set of n items.
In this case, our binomial is (d-5y), and the exponent is 6. We can use Pascal's triangle to find the coefficients for each term in the expansion:
Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1
Row 6: 1 6 15 20 15 6 1
By using the numbers in the 6th row of Pascal's triangle, we can write out the expanded form of the binomial:
(d-5y)^6 = C(6, 0)*d^6*(-5y)^0 + C(6, 1)*d^5*(-5y)^1 + C(6, 2)*d^4*(-5y)^2 + C(6, 3)*d^3*(-5y)^3 + C(6, 4)*d^2*(-5y)^4 + C(6, 5)*d^1*(-5y)^5 + C(6, 6)*d^0*(-5y)^6
Simplifying each term, we have:
(d-5y)^6 = 1*d^6*(-5y)^0 + 6*d^5*(-5y)^1 + 15*d^4*(-5y)^2 + 20*d^3*(-5y)^3 + 15*d^2*(-5y)^4 + 6*d^1*(-5y)^5 + 1*d^0*(-5y)^6
(d-5y)^6 = d^6 + 6*(-5)*d^5*y + 15*25*d^4*y^2 + 20*(-125)*d^3*y^3 + 15*625*d^2*y^4 + 6*(-3125)*d*y^5 + (-15625)*y^6
Therefore, the expanded form of (d-5y)^6 is:
d^6 - 30d^5y + 375d^4y^2 - 2500d^3y^3 + 9375d^2y^4 - 18750dy^5 + 15625y^6