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Combinatorial proof of Negative Binomial Identity $(1+x)^{-n}= \sum_{k=0}^{\infty} (-1)^k \binom{n+k-1}{k} x^k$

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#combinatorics #permutations #binomial-coefficients #binomial-theorem
130
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Show $\left(1-\frac x k\right)^k<\left(1-\frac {x}{ k+1}\right)^{k+1}$

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#calculus #inequality #binomial-theorem
83
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In how many ways a student can get $2m $ Marks

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#combinatorics #polynomials #binomial-coefficients #binomial-theorem
473
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How to solve interest problem without using a crazy binomial expansion?

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#statistics #finance #binomial-theorem #actuarial-science
369
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Proof of Inequality by Binomial Theorem

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#real-analysis #binomial-theorem
261
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Which type of random walk has distribution of "scaled" binomial coefficient?

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#probability #combinatorics #probability-theory #probability-distributions #binomial-theorem
69
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Polynomial of $n-1$ degree

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#polynomials #complex-numbers #binomial-theorem
147
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A Subtle Point in Rudin's Proof on $e$

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#real-analysis #limits #proof-explanation #binomial-theorem
73
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Writing $(m+2)^n-(m-2)^n$ in summation notation.

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#calculus #algebra-precalculus #summation #binomial-theorem
967
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Sum of weighted binomial coefficients

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#combinatorics #binomial-coefficients #binomial-theorem
64
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Generalizing $(^4C_0)^2-(^4C_1)^2+(^4C_2)^2-(^4C_3)^2+(^4C_4)^2=\,^4C_2$

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#binomial-coefficients #binomial-theorem
50
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What kind of expansion is this?

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#binomial-coefficients #binomial-theorem
1.4k
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A lower bound for $(1-e^x)^n$

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#calculus #inequality #exponential-function #binomial-theorem #upper-lower-bounds
63
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Could someone help clarify the meaning of this pre-calculus question on Binomial Expansion?

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#algebra-precalculus #binomial-coefficients #binomial-theorem
116
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How to prove the two identities

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#abstract-algebra #summation #binomial-theorem

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