Questions tagged [binomial-theorem]
For questions related to the binomial theorem, which describes the algebraic expansion of powers of binomials.
2,540 questions
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Binomial sum without using sterling numbers
I came across this binomial sum-
$p+q=99^{98}(1+99^{2})-\frac{99 \cdot 98^{98}}{1}(1+98^{2})+\frac{99 \cdot 98 \cdot 97^{98}}{1 \cdot 2}(1+97^{2})-\frac{99 \cdot 98 \cdot 97\cdot 96^{98}}{1 \cdot 2 \...
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Is it possible to simplify a sum of these two cube roots by completing the cube?
Let $x$ be a real number. Consider the following expression:
$$
\sqrt[3]{\frac{x^{3} - 3x + \left(x^{2} - 1\right)\sqrt{x^{2} - 4}}{2}} + \sqrt[3]{\frac{x^{3} - 3x - \left(x^{2} - 1\right)\sqrt{x^{2} -...
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In the expansion of $(1 + x + x^2 + \cdots + x^{10})^3$, what is the coefficient of: $x^5$ [duplicate]
Using multinomial theorem the powers of terms should sum to $a + 2b + 3c + 4d + 5e = 5$;
We get following solutions $e = 1$;
$a = 1, d = 1$;
$b = 1, c = 1$;
$c = 1, a = 2$;
$a = 1, b = 2$;
So we get $\...
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1
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A finite sum formula for $\sin^k z$ for $z\in\mathbb{C}$, $k\in\mathbb{N}$
I am reading this article by Zudilin. In page $523$, there is a formula for $a,b$ odd positive integers, $n$ a positive integer and $\tau$ a complex number,
$$\sin^{2m+b}\pi n\tau=\chi_m \frac{(-1)^{(...
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Combinatorial interpretation of the sign-alternating binomial formula
The binomial formula $(x+y)^{n}=\sum_{k=0}^{n}\binom{n}{k}x^{k}y^{n-k}$ could
be interpreted, when $x,y$ are positive integers, as a two-way counting of
"words" of length $n$ which use ...
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Asymptotic bound on modified binomial sum
Problem statement
For all $x>0$, $n, m \in \mathbb{N}$, let
$$
S(x, n, m) \triangleq \sum_{j=0}^n \binom{n}{j} (1 - x)^{n-j} x^j j^3 \frac{1}{1 + \frac{j}{m} }
$$
Assuming $n = o(m)$, for all $c&...
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Finding $\sum_{r=0}^n \sum_{s=0}^n \frac{(-1)^{r+s}}{{n \choose r}\left({n \choose s}+ {n \choose r}\right)}$
Let $S_n=\sum_{r=0}^n \sum_{s=0}^n \frac{(-1)^{r+s}}{{n \choose r}\left({n \choose s}+ {n \choose r}\right)}.$
Interchange $r$ and $s$, to have
$S_n=\sum_{s=0}^n \sum_{r=0}^n \frac{(-1)^{s+r}}{{n \...
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Evaluate $\sum_{r=1}^{101}(-1)^{r-1}(1+\frac12+\frac13+...+\frac1r){101\choose r}$
Evaluate $\sum_{r=1}^{101}(-1)^{r-1}(1+\frac12+\frac13+...+\frac1r){101\choose r}$
My Attempt:
If $r$ was starting from $0$, I could have written $(1-1)^{101}$.
Now, maybe ${101\choose r}$ can be ...
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2
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Asymptotics of $(x+a)^{(x+a)}$ for large $x$ [closed]
According to Mathematica, the $x \to \infty$ asymptotics of $f(x) = (x+a)^{(x+a)}$, for $a>0$, are given my
$$f(x)\sim x^{(a+x)}e^{a+\frac{a^2}{2x}}.$$
How is this shown?
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Binomial identity $ \sum_{k=0}^{n} (-1)^k \binom{n}{k} \binom{2n - 2k}{n+1} = n 2^{n-1}$ [duplicate]
Prove: $$ \sum_{k=0}^{n} (-1)^k \binom{n}{k} \binom{2n - 2k}{n+1} = n 2^{n-1}. $$
Combinatorially, both sides count the number of subsets $X \subseteq A:=A_1\sqcup A_2\sqcup\ldots \sqcup A_n$ such ...
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Is there any specific formula for ($a^{n}+b^{n}$) [closed]
I was just wondering if there is any specific formula for $a^{m}+b^{m}$.
I have noticed that if we have $a^n+b^n$, and $n$ is odd, then we have :
$a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b^1+a^{n-3}b^2-...-ab^{n-...
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1
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A property of positive real numbers
Suppose $0\leq a\leq 1$ and $b\geq 1, c\geq 1$ are any real numbers. Then may I know if there is any result in the literature as given below:
$$\left({a+\sqrt[n]{b}}\right)^{n}\left({a+{c}}\right)\...
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Clarification of Part of a Binomial Expansion from my textbook for $(1+1/n)^n$
My textbook states that:
$$\sum_{i=0}^n\binom ni \left(\frac1n\right)^n$$
is equal to
$$\sum_{i=0}^n\frac{n(n-1)\cdots(n-1+1)}{i!}\left(\frac1n\right)^n$$
But seeing as (n choose i) is equal to (n!/(i!...
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Summing $\sum_{r=0}^{[n/2]} {n-r \choose r} x^r$
While working at Mathematica, I realized that $$S_m=\sum_{r=0}^{m} {n-r \choose r} x^r \tag{1}$$ is sum of two hypergeometric series when $m=n$, but when $m=[n/2]$,
$S_{n/2}$ can be expressed as
$$S_{...
- 47.2k
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Evaluate ${^5C_1}{^5C_5}-{^5C_2}{^{10}C_5}+{^5C_3}{^{15}C_5}-{^5C_4}{^{20}C_5}+{^5C_5}{^{25}C_5}$
Evaluate ${^5C_1}{^5C_5}-{^5C_2}{^{10}C_5}+{^5C_3}{^{15}C_5}-{^5C_4}{^{20}C_5}+{^5C_5}{^{25}C_5}$
My Attempt:
We are essentially looking for the coefficient of $x^5$ in ${^5C_1}(1+x)^5-{^5C_2}(1+x)^{...
- 10.8k
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1
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If $f(x)=\sum_{r=0}^{20}(-1)^{r}{20\choose r}\left(1+3\sin x\right)^{20-r}$ then evaluate $\sum_{r=0}^{\infty}\left(f(\frac{\pi}{6})\right)^r$
If $$f(x)=\sum_{r=0}^{20}(-1)^{r}{20\choose r}\left(1+3\sin x\right)^{20-r}$$ then evaluate $$\sum_{r=0}^{\infty}\left(f\left(\dfrac{\pi}{6}\right)\right)^r$$
My solution:
I obtained $f(x)=\left(1+3\...
- 4,708
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4
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When $f(n)$ divided by $5$ then which one of the following is true about the remainder obtained?
The following question is taken from the practice set of JEE.
Question:
Consider
$f(n)={^{2n+1}C_1}+{^{2n+1}C_3}2^3+{^{2n+1}C_5}2^6+{^{2n+1}C_7}2^9+...+{^{2n+1}C_{2n+1}}2^{3n}$
When $f(n)$ divided by $...
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The value of ${20\choose7}-{20\choose8}+......-{20\choose20}$ is equal to [duplicate]
The value of ${20\choose7}-{20\choose8}+......-{20\choose20}$ is equal to
(a) ${19\choose 13}$
(b) ${19\choose 14}$
(c) ${20\choose 13}$
(d) None of these
My solution:
$(1+x)^{20}={20\choose 0}+{20\...
- 4,708
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Direct proof involving binomial theorem is wrong, but I don't understand why. [closed]
I have to prove:
$$ \binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \cdots + \binom{n}{n} = 2^n $$
L.H.S.
$$ \frac{n!}{0!(n-0)!} + \frac{n!}{1!(n-1)!} + \frac{n!}{2!(n-2)!} + \cdots + \frac{n!}{n!(n-n)!} ...
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Showing $\binom{n}{k} =\frac{n(n−1)···(n−k+1)}{k!}$ [closed]
Just one line in Burton's number theory has made me waste hours and facepalm myself in shame.
Here it is:
$$\binom{n}{k} =\frac{n(n−1)···(k+1)}{(n-k)!}=
\frac{n(n−1)···(n−k+1)}{k!}$$
First ...
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Binomial coefficient relation for rational powers
In Eckle's book "Models of Quantum Matter" he gives the exercise to show that the two expressions (7.14)
and (7.230)
are the same (note $w_1 = (u_1,v_1),\, w_2 = (u_2, v_2)$). Could anyone ...
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Pi using Newton's approximation
Instead of taking the angle that the revolving line makes with the x-axis to be 60 degrees, if we take it to be 75 degrees, it should converge faster using the expansion for (1 - x^2)^0.5. But, when I ...
user1385195
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Proof verification : $1+p\in (\mathbb Z/p^n\mathbb Z)^\times$ and its order is $p^{n-1}$
I want to show that $1+p$ is an element of $(\mathbb Z/p^n\mathbb Z)^\times$ and its order is $p^{n-1}$ for odd prime $p$. (Dummit/Foote 2.3.21)
The textbook said one can prove it by using the ...
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Fractional Pascal's Triangle [closed]
Is there a fractional version of Pascal's triangle for binomial expansion similar to the standard triangle used for binomial expansion? If so, is it related to the Gamma function?
\begin{array}{cccccc}...
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Computing $ \binom{n-k}{i} i = (n-k) \binom{n-k-1}{i-1}$
$\sum_{i=0}^{n-k} \binom{n-k}{i} i (n-1)^{n-k-1-i}$
This sum was simplified in a paper I was reading with the identitiy: $ \binom{n-k}{i} i = (n-k) \binom{n-k-1}{i-1}$ and I simply cannot find this ...
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