# Questions tagged [binomial-theorem]

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

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### How do you find the coefficient of $x$ in $(x + 1)^2$?

I want to learn how can I find out the coefficient of the variable $x$ in the expression $(x + 1)^2$. It is a case of a perfect square expansion.
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### approximated value Using Binomial theorem Expansion

Find An approximated value for 4√630 , (2.9)5 Using (i) Binomial theorem Expansion (ii) Differentiation Methods enter image description here
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### How may one solve problems over expressions like $(2+px)^6$ without the binomial theorem?

A friend of mine posed a problem on a mathematics discord server. The coefficient of the $x^2$ term in the expansion of $(2+px)^6$ is $60$. Find the value of the positive constant $p$. I ...
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### calculate the given sum (related to Newton's binomial formula) [duplicate]

i was given the following question: given n is a positve integer, calculate the given sum: $${n \choose 1} -2{n \choose 2} +3{n \choose 3}+ \ldots +(-1)^{n-1}\cdot n {n \choose n}$$ I looked at ...
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### Sum of product of binomials

While working on a combinatorics problem, I found that this result had to be true: $$\sum_{i=0}^n\frac{(a-b)^i(b-c)^{n-i}}{i!(n-i)!}=\frac{(a-c)^n}{n!}$$ for $a\geq b\geq c$, with $a,b,c\in\Bbb N$. ...
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### Series Expansion and Big-Oh Notation: Expanding $\sqrt{x^2+\mathcal{O}(x^3)}$ to get $x+\mathcal{O}(x^{3/2})$

I've been reading through a thesis and trying to rederive all of the equations used. However, I've come across an expansion that I'm unsure on. The following is a simplified form of two consecutive ...
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### Confused about the meaning behind making x = 2 in the binomial theorem

So the binomial theorem states $(1+x)^n = \sum^{n}_{k=0}$$n \choose k$$x ^ k$. Now I understand that each term of the sum represents the number of ways to arrange 1 and $x$ out of $n$ choices, so ...
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### May I know why is the fact that P(The best is among the first n) is calculated in this way?

I am trying to solve this problem: On the basis of an interview, the N candidates for admission to a college are ranked in order according to their mathematical potential. The candidates are ...
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### binomial theorem combinatorics consideration question [duplicate]

i have an identity that i need to prove in 2 ways $$\sum_{k=m}^n \binom k m = \binom {n+1} {m+1}$$ the first way was to prove using induction which wasn't a problem since n≥m so just do the base and ...
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### Fractional exponent for binomial theorem

If I am trying to expand $(a+b)^{\frac{2}{3}}$, can I use the binomial theorem like so: $$\sum_{k=0}^{\frac{2}{3}}{\frac{2}{3}\choose k}a^{\frac{2}{3}-k}b^k$$ or will that not work, since the last ...
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### $q$-analogue of $\sum_{k=0}^n \, {n \choose k} = 2^n$

Is there a $q$-analogue of the formula $\sum_{k=0}^n \, {n \choose k} = 2^n$ in terms of the $q$-binomial coefficient ${n \choose k}_q$ and $(2^n)_q=(1+q)...(1+q^n)$?
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### Proving $(1+x)f'(x)=a f(x)$ where $f(x)=\sum_{n=0}^\infty {a\choose n}x^n$

A function $f$ is defined for $-1<x<1$ by $f(x)=\sum_{n=0}^\infty {a\choose n}x^n$. Here $a$ is a real number which is neither zero nor a positive integer. Prove that $(1+x)f'(x)=a f(x)$. I took ...
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### Proof of binomial theorem for non-integers

Firstly, yes, I know how to expand expressions like $(a+b)^{({1}/{2})}$ but I want a rigorous proof of why it is okay to replace binomial coefficients like ${1/2}\choose{2}$ with $({1/2})({1/2-1}))/2$...
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### Prove $\binom{n}{k_1,…,k_m} = \sum_{i=1}^m \binom{n - 1}{k_1,…,k_{i - 1},..,k_m}$

I have a question ask to prove $$\binom{n}{k_1,...,k_m} = \sum_{i= 1}^m \binom{n - 1}{k_1,...,k_{i - 1},..,k_m}$$ I'm not sure how to approach this question, but the only thing that I noticed the LHS ...
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### Factorise $x^4+y^4+(x+y)^4$

I think this problem involves making use of symmetry in some way but I don't know how. I expanded the $(x+y)^4$ term but it din't help in the factorization. I am very bad at factorizing, so I didn't ...
### Find coefficients of $x^2$ and $x^3$ in the expansion of $(3 − 2x) ^6$
The coefficients of $x^2$ and $x^3$ in the expansion of $(3 − 2x) ^6$ are $a$ and $b$ respectively. What is the value of $a$ and $b$? They need to be good at the binomial theorem and know the ...