TEAM PROJECT. Permutations, Combinations. (a) Prove Theorem 2. (b) Prove the last
Chapter 24, Problem 24.1.78(choose chapter or problem)
TEAM PROJECT. Permutations, Combinations.
(a) Prove Theorem 2.
(b) Prove the last statement of Theorem 3.
(e) Derive (11) from (8).
(d) By the binomial theorem,
\((a+b)^{n}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) a^{k} b^{n-k} \),
so that \(a^{k}b^{n - k}\) has the coefficient \((_{k}^{n})\). Can you conclude this from Theorem 3 or is this a mere
coincidence?
(e) Prove (14) by using the binomial theorem.
(f) Collect further formulas for binomial coefficients from the literature and illustrate them numerically.
(a + b)^n = sum ^n _ k = 0 (_k^n) a^k b^n-k
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