How do you use the Binomial Theorem to expand #(d+3)^7#?

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Answer 1 · 2016-07-04T19:11:29

#= d^7 + 21d^6 + 189d^5 + 945d^4 + 2835d^3 + 5103d^2 + 5103d + 2187#

#(x+y)^n = sum_(k=0)^n ((n),(k))x^(n-k)y^k#

where #((n),(k)) = (n!)/(k!(n-k)!)# taking note that #0! = 1#

#(d+3)^7 = sum_(k=0)^7 ((7),(k))d^(7-k)3^k#

#= ((7),(0))d^(7)3^0 + ((7),(1))d^(6)3^1+((7),(2))d^(5)3^2... ((7),(7))d^(0)3^7#

#= (7!)/(0!7!)d^7 + (7!)/(1!6!)d^(6)*3 + (7!)/(2!5!)d^5*9 ... (7!)/(7!0!)*3^7#

#= d^7 + 21d^6 + 189d^5 + 945d^4 + 2835d^3 + 5103d^2 + 5103d + 2187#

Note, for brevity I have skipped out a chunk of the calculation but the method is the exact same so you should have no issue with reproducing it