Derive the binomial expansion and connect algebraic and combinatorial identities.
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Try
Expanding (a+b)³ means choosing a or b from each of three factors. The term a²b appears three times because exactly one factor supplies b. Coefficients record choices, not arbitrary numbers introduced by symbolic manipulation.
Predict
Predict the coefficient of aⁿ⁻ᵏbᵏ in n factors. Choose the k positions supplying b: there are C(n,k) choices. The binomial theorem groups the entire expansion by k. Here n is a nonnegative integer; infinite generalized binomial series require a separate theory.
Experiment
For four factors, draw or list the six two-position subsets that supply b². Verify the coefficient of a²b² is six. Then replace a with 2 and b with x; the coefficient of x² gains a factor 2² from the two constant selections.
Understand
Worked identity: count k-subsets of an n-element set by a distinguished element z. Those excluding z contribute C(n−1,k); those including it choose their remaining k−1 members in C(n−1,k−1) ways. The classes are disjoint and exhaustive, proving Pascal’s recurrence for the usual interior indices.
The row sum ΣC(n,k)=2ⁿ follows either by evaluating (1+x)ⁿ at x=1 or by counting all subsets by their sizes. Complementation proves symmetry C(n,k)=C(n,n−k), connecting the two sides of Pascal’s triangle.
Invalid expansion: (a+b)ⁿ=aⁿ+bⁿ. It discards mixed choices; at n=2 the missing terms total 2ab.
Apply
In coefficient problems, identify how many factors supply each term and include any constant powers. In a proof of a combinatorial identity, describe the same objects counted in two ways rather than merely presenting matching numerical rows.
Continue
Depth: differentiating a polynomial expansion yields weighted sums such as ΣkC(n,k)=n2ⁿ⁻¹ for n≥1; a purely combinatorial proof counts a subset with one distinguished member. Two different proofs can reveal different meanings of the same identity.
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Practice and reasoning
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Find the coefficient of x³ in (1+x)⁶.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Find the coefficient of x² in (2+x)⁴.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which identity is Pascal's recurrence?
The sum of all coefficients of (1+x)ⁿ is:
Prove (a+b)ⁿ=Σ(k=0..n) C(n,k)aⁿ⁻ᵏbᵏ for nonnegative integer n.
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Give a bijective proof of C(n,k)=C(n,n−k), rather than canceling factorials.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
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