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What is the sum of the numerical coefficients (including the constant term) of the expanded form of the expression below? (t−2)6(t-2)^6

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Full solution

Q. What is the sum of the numerical coefficients (including the constant term) of the expanded form of the expression below? (t−2)6(t-2)^6
  1. Apply Binomial Theorem: Use the Binomial Theorem to expand (t−2)6(t-2)^6. The Binomial Theorem states that (a−b)n=∑k=0n(nk)⋅a(n−k)⋅(−b)k(a-b)^n = \sum_{k=0}^{n} \binom{n}{k} \cdot a^{(n-k)} \cdot (-b)^k.
  2. Calculate Coefficients: Calculate the coefficients for each term in the expansion.\newlineThe coefficients will be determined by ((6k))(6 \choose k) for k=0k=0 to 66.
  3. List Coefficients: List out the coefficients.\newline((60)=1)(6 \choose 0) = 1, ((61)=6)(6 \choose 1) = 6, ((62)=15)(6 \choose 2) = 15, ((63)=20)(6 \choose 3) = 20, ((64)=15)(6 \choose 4) = 15, ((65)=6)(6 \choose 5) = 6, ((66)=1)(6 \choose 6) = 1.
  4. Add Coefficients: Add up the coefficients.\newlineSum = 1+6+15+20+15+6+11 + 6 + 15 + 20 + 15 + 6 + 1.
  5. Perform Addition: Perform the addition.\newlineSum = 6464.

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