# Questions tagged [binomial-theorem]

For questions related to the binomial theorem, which describes the algebraic expansion of powers of binomials.

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### AB is a chord of length 2ka of a circle of radius a. The tangents to the circle at A and B meet in C if k^7 is negligible calculate the area of ABC

AB is a chord of length 2ka of a circle of radius a. The tangents to the circle at A and B meet in C. Show that, if k is so small compared with unity that $k^7$ is negligible, the area of the triangle ...
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### Can we solve this only by using bijection?

The question is : The number of possible outcomes in a throw of n ordinary dice in which at least once of the dice shows an odd number are: Now, we can simply apply bijection principle , and ...
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### Confusion regarding the nth term in a binomial expansion (positive integral index)

The expansion for a binomial is - $(a+b)^n=\sum_{k=0}^n{n\choose k}a^{n-k}b^{k}$ But it could also be this $(a+b)^n=\sum_{k=0}^n{n\choose k}a^kb^{n-k}$ Both of the expansions are correct because the ...
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### Binomial coefficient identity with alternating summands [closed]

Is there a nice closed form for $\sum_{k=0}^{l} (-1)^k \binom{n}{k} \binom{n-k}{l-k}.$ I feel like there must be but I cannot find it.
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### Finding $\sum_{k=0}^n (-1)^k A_k {n \choose k}, ~\text{if}~ (1+x+x^2)^n=\sum_{k=0}^{2n} A_k x^k$
Finding $S=\sum_{k=0}^n (-1)^k A_k{n\choose k}$, if $(1+x+x^2)^n=\sum_{k=0}^{2n} A_k x^k.$ \begin{align} (1+x+x^2)^n &= A_0+A_1x+A_2x^2+\dots+A_nx^n+\dots+A_{2n}x^{2n} \\ (1-1/x)^n &= {n \...