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Find the coefficients in the expansions of (a + b)11 and (a + b)12.
Hint: The coefficients of (a + b)11 form the row 1 11 55 ... 55 11 1 of Pascal's triangle, each term of which is the sum of the two terms above it in the previous row. The coefficients of (a + b)12 form the row 1 12 66 ... 66 12 1, which can be obtained similarly from the previous row.
Express the sum of the terms in the nth row of Pascal's triangle as an algebraic identity.
Hint: Use summation notation, with a binomial coefficient as the summand.
Prove the identity for the sum of the nth row of Pascal's triangle.
Hint: Consider the expansion of (a + b)n as it pertains to the identity, and choose some specific values for a and b.
Arrange the first few trinomial coefficients geometrically so that they satisfy an additive relation.
Hint: What do the coefficients of the nth "row" add up to?
Find a formula for trinomial coefficients, and prove that it satisfies the additive relation.
Hint: Reduce trinomial expansion to the case where we already have an explicit formula for the coefficients — binomial expansion.
Find an additive relation for d-nomial coefficients.
Hint: Generalize from known cases.
Find an explicit formula for d-nomial coefficients, and prove that it satisfies the additive relation.
Hint: Generalize from known cases.
Find a relationship between the rows of d-nomial coefficients and the rows of (d + 1)-nomial coefficients.
Hint: To start, look for resemblances to Pascal's triangle in the rows of Pascal's tetrahedron and resemblances to the tetrahedron in the tetrahedral cross sections of the pentatope.