Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

learn more… | top users | synonyms (1)

2
votes
1answer
37 views

Inequality for an alternating sum of binomial coefficients times a fraction

I need to show that $$\sum_{n=0}^x(-1)^n {x\choose n}\left(\frac{1}{n+l-1}-\frac{1}{n+l}\right) >0,$$ for any $l>0$. I tried to prove this, but I didn't get anywhere. Apparently the above ...
4
votes
4answers
57 views

binomials product alternating sum calculation

I need to somehow prove that $\sum\limits_{k = 0}^{n - 1} {n \choose k} {3 n - k - 1 \choose 2 n - k}(-1)^k = (-1)^{n + 1} {2 n - 1 \choose n}$. I didn't manage to do it using induction or any ...
0
votes
2answers
29 views

$\binom{n}{0}-\frac{1}{(2+\sqrt3)^2}\binom{n}{2}+\frac{1}{(2+\sqrt3)^4}\binom{n}{4}-\frac{1}{(2+\sqrt3)^6}\binom{n}{6}+…=$

If $n=12m,m\in N$,prove that $\binom{n}{0}-\frac{1}{(2+\sqrt3)^2}\binom{n}{2}+\frac{1}{(2+\sqrt3)^4}\binom{n}{4}-\frac{1}{(2+\sqrt3)^6}\binom{n}{6}+.....=(-1)^m\left(\frac{2\sqrt2}{1+\sqrt3}\right)^n$ ...
1
vote
0answers
41 views

Prove the following result on binomial coefficients

If $(1+x)^n=^n\!\!C_0+^n\!\!C_1x+^n\!\!C_2x^2+\cdots+^n\!\!C_nx^n$, then show that $$(^n\!\!C_0-^n\!\!C_2+^n\!\!C_4-^n\!\!C_6+\cdots)^2+(^n\!C_1-^n\!\!C_3+^n\!\!C_5-\cdots)^2\\ ...
6
votes
2answers
84 views

Rank of a matrix of binomial coefficients

This question arose as a side computation on error correcting codes. Let $k$, $r$ be positive integers such that $2k-1 \leqslant r$ and let $p$ a prime number such that $r < p$. I would like to ...
-4
votes
1answer
83 views

President Obama proposed the elimination of taxes on dividends paid to shareholders on the grounds that they result in double taxation.

President Obama proposed the elimination of taxes on dividends paid to shareholders on the grounds that they result in double taxation. The earnings used to pay dividends are already taxed to the ...
0
votes
2answers
56 views

Number of ways to put $n$ red cards and $k$ black cards into $4$ distinct jars so that every jar has a card.

So if we define two functions $f_1 [n]\rightarrow [4]$ and $f_2[k]\rightarrow [4]$, in order to do this problem we need for the functions to be onto. This is simple enough, right? If $f_1$ is onto ...
1
vote
3answers
39 views

How can I solve for N in the inequality below? [closed]

I know that the answer is N = 537, but I'm not sure how to solve for N analytically. More precisely, my question is how can I simplify the sum of the binomial coefficients ? The inequality is at ...
7
votes
3answers
159 views

Prove that $2^{n-1}$ divides $\binom{n}{1} + \binom{n}{3}5 + \binom{n}{5}25 + \binom{n}{7}125 + \cdots$ for $n \geqslant 1$.

Prove that $2^{n-1}$ divides $\binom{n}{1} + \binom{n}{3}5 + \binom{n}{5}25 + \binom{n}{7}125 + \cdots$ for $n \geqslant 1$. Assume $\binom{n}{k} = 0$ if $k>n$. Does anyone know an elementary ...
-1
votes
2answers
26 views

Power of prime P in $nCr$ [closed]

How can we determine power of prime $P$ in $C(n,r)$, where $C(n,r)$ denotes the number of combination of $n$ objects taken $r$ at a time. (Also denoted $\binom{n}{r}$ and called a binomial ...
4
votes
2answers
165 views

Seeking a combinatorial proof $\sum _{k=0}^n (n-2k)^2\binom{n}{k}=n\times 2^n$

I would appreciate if somebody could help me with the following problem Q: Seeking a combinatorial proof $(\binom{n}{k}=\frac{n!}{k! (n-k)!} )$ $$\sum _{k=0}^n (n-2 k)^2 \binom{n}{k}=n\times 2^n$$
1
vote
1answer
47 views

Finite sum and limit invoving binomial coefficient

I have found some interesting formulas involving binomial coefficients with the help of Mathematica. But I have no idea how it did. Could anyone help me? Here they are: $$\sum_{k=0}^{m-1} 2^{-2k} ...
3
votes
0answers
89 views

Square of hockey stick identity: $\sum_{i=r}^n{i \choose r}^2$

Evaluate $\sum_{i=r}^n{i \choose r}^2$ where $n,r\in \mathbb{N},n>r$. This looks like the hockey stick identity but I can't find a way to evaluate it without a computer. Can someone help me out?
2
votes
1answer
100 views

Prime dividing binomial coefficient involving prime power

I was wondering if there was a straightforward proof of the following fact (which I can show is true for specific cases, but not generally): Let $n$ be composite, and let $p$ be a prime factor $n$. ...
0
votes
2answers
106 views

How do I prove this using mathematical induction? [duplicate]

$\sum_{k = 1}^{n}k\binom{n}{k}=n2^{n-1}$ How do I prove this using mathematical induction?
5
votes
2answers
108 views

Prove that $\sum_{r=1}^n \frac 1{r}\binom{n}{r} = \sum_{r=1}^n \frac 1{r}(2^r - 1)$

Prove that $\sum_{r=1}^n \frac 1{r}\binom{n}{r} = \sum_{r=1}^n \frac 1{r}(2^r - 1)$ One thing I have tried is to represent both $\binom{n}{r}$ and $2^r$ as sums of binomial coefficients, i.e. ...
2
votes
1answer
63 views

is sub-matrices of a pascal matrix non-singular?

I have a regular n*n symmetric Pascal matrix, I choose a square inside this matrix and I want to know if the sub-matrix is non-singular?
-2
votes
1answer
80 views

Combinatorial proof of $\sum_{j=i}^n { n \choose j} { j \choose i} (-1)^{n-j}=0$

I would like a combinatorial proof using committee and subcommittee selection for the following identity. $$\sum_{j=i}^n { n \choose j} { j \choose i} (-1)^{n-j}=0$$ with $i<n$. This is from ...
3
votes
1answer
260 views

Finding the number of lattice paths

Find the number of lattice path of length $2n$ that starts on $(0, 0)$ such that for all the points $(x, y)$ in the path, $x < y$. So pretty much all the points besides the origin are strictly ...
0
votes
1answer
26 views

If I have a 45 bit ternary number, where x bits must be 0, y bits must be 1, and z bits must be 2…

... How, knowing a specific x, y, and z, can I find how many different combinations of 0,1,and 2 can I have? I have a specific problem. 15 bits must be zero, 20 must be one, and 10 must be 2. But I ...
0
votes
1answer
37 views

Simplify an Algebraic Expression

I would like to show the following algebraic equality holds for every $N \in \mathbb{N}$, $$ \frac{1}{N}\sum\limits_{k = 0}^N \frac{N!}{k!(N-k)!}{{\theta^k}{{\left( {1 - \theta} \right)}^{N - ...
1
vote
1answer
34 views

Partial sum of coefficients of polynomials

Let me define polynomials of form $1+x^2+x^3+\cdots+x^k$ as $P(k,x)$. Let $$Q(x)=\prod_{k=1}^{n}P(k,x)$$ How can I find the sum of coefficients for which exponent of $x$ is $\le T$, where $0 \le T ...
2
votes
2answers
209 views

Binomial Theorem Question (Expansion of Three Terms)

I have the term: $(1 + 2x - x^2)^4.$ The question asks me to find the coefficient of $x^5$. My solution: $\sum\limits_{i=0}^4 {4 \choose r} (1)^{4-r}(2x-x^2)^r$ I then factored out x from ...
4
votes
0answers
328 views

LCM of binomial coefficients and related functions

I know about the following identity: $$\displaystyle \text{lcm} \left( {n \choose 0}, {n \choose 1}, ... {n \choose n} \right) = \frac{\text{lcm}(1, 2, ... n+1)}{n+1}$$ 1) Is there any method to find ...
4
votes
4answers
167 views

Li Shanlan's combinatorial identities

I am struggling to prove the following combinatorial identities: $$(1)\quad\sum_{r=0}^m \binom{m}{r}\binom{n}{r}\binom{p+r}{m+n} = \binom{p}{m}\binom{p}{n},\quad \forall n\in\mathbb N,p\ge m,n$$ ...
4
votes
3answers
132 views

Is $72!/36! -1$ divisible by 73?

Is $\frac{72!}{36!}-1$ divisible by the number 73? I am not getting a clue in which direction should I go, though I did small amount of work by converting the above expression in the below given form ...
-3
votes
1answer
44 views

Prove that (n + 1) (n choose m) = (n + 1 − m) ((n + 1) choose m) [closed]

Let m and n be integers with $0 ≤ m ≤ n$. There are $n + 1$ students in Carleton’s Computer Science program. The Carleton Computer Science Society has a Board of Directors, consisting of one president ...
4
votes
1answer
64 views

Prove $\sum_{k=0}^n\binom{2n+1}{2k}=4^n$

I once had to show that $\cos(x)\sin(x)=\frac{1}{2}\sin(2x)$ using the Cauchy product and relied on $$\sum_{k=0}^n\binom{2n+1}{2k}=4^n.$$ However I never came up with a proof why this is true - is ...
2
votes
1answer
38 views

show that $\sum\limits_{k=0}^n \binom{n}{k}2^k=\sum\limits_{k=0}^n \binom{n}{k}2^k\cdot 1^{n-k}= 3^n$

Show that $$\sum\limits_{k=0}^n \binom{n}{k}2^k=\sum\limits_{k=0}^n \binom{n}{k}2^k\cdot 1^{n-k}= 3^n$$ I know this true but i really having a hard time arrive there. Is it just that $(2+1)^n = ...
2
votes
2answers
60 views

Binomial coefficients inequation problem

Can anyone help me solve this: $$5\binom{13}{x} < \binom{x + 2}{4}$$ After turning it to factorial I don't know what to do nothing seems to cancel out. $x$ is a positive integer. I end up with this ...
1
vote
3answers
25 views

Calculating limit involving binomial coefficient

$$\lim_{n\rightarrow \infty} \binom{n}{k} h^{(n-k)} , |h| < 1$$ I'm trying to evaluate this limit, but like the $h$ messes me up each time. What I'm doing is trying to prove that the infinite ...
1
vote
1answer
63 views

Binomial transform of Catalan numbers formula

How to prove that OEIS A007317 Binomial transform of Catalan numbers $a_{n}: 1, 2, 5, 15, 51, 188, 731, 2950, 12235, 51822, .. (n = 1, 2, ..)$ has a recurrence formula: $(n+2)a_{n+2} = (6n+4)a_{n+1} - ...
2
votes
5answers
80 views

What is the coefficient of $x^3 y^4$ in the expansion of $ (2x-y+5)^8$

I was thinking of doing $\binom{8}{4}$ but not sure if right.
0
votes
1answer
32 views

Find the coefficient of the given term when (u^2 - v^2 ) ^10 expanded by the binomial theorem?

The term is u^16 v^4 When (u^2 - v^2 ) ^10 is exanded by the binomial theorem. My book uses Combinations, but I'm not sure if it works if u and v are squared?
5
votes
2answers
104 views

How do we prove the following binomial identity?

I tried to prove it by expanding the left hand side, but to no avail. Can you please explain me how to prove this statement? I'm thinking calculus(differentiation) can be used to prove this, as ...
2
votes
1answer
85 views

Help Show Binomial Identity: $\sum_{j=0}^{n} {n \choose j}{m+j \choose n} = \sum_{j=0}^{n} {n \choose j}{m \choose j}2^j$ [duplicate]

I have been trying to solve this problem that I found in my old course notes for some time, but I have not been successful. Can anyone suggest a strategy or provide a hint? ...
-2
votes
3answers
75 views

Trying to prove $\sum\limits_{k=0}^{n}\binom{n}{k}=(1+1)^n$ [duplicate]

I am trying to show in the following equality that the left hand side equals the right hand side. I tried expanding out the summation but that didn't get me anywhere. Could somebody provide a hint? ...
6
votes
1answer
67 views

$\frac{2n\choose n}{n+2}\not\in\mathbb N$ and $n\neq3k+1$ and $n\neq4k+2$

Are there any natural numbers $n\not\equiv1\bmod3$, and $n\not\equiv2\bmod4$, so that $~\dfrac{\displaystyle{2n\choose n}}{n+2}\not\in\mathbb N$ ? Since $C_n=\dfrac{\displaystyle{2n\choose ...
3
votes
2answers
71 views

$\binom nk=\sum_{j=0}^{\lfloor\frac k2\rfloor}(-1)^j\binom nj\binom{n+k-2j-1}{n-1}$

Prove combinatorially (using inclusion-exclusion) that$ \binom nk=\sum_{j=0}^{\lfloor\frac k2\rfloor}(-1)^j\binom nj\binom{n+k-2j-1}{n-1}$ Hi, everyone. I'm at a loss here. I've been trying ...
1
vote
1answer
36 views

Divisibility test using perhaps binomial thorem

I have to determine if $17^{21} + 19^{21}$ is divisible by any of the following numbers (a) 36 (b) 19 (c) 17 (d) 21. I am trying to find using binomial expansion but getting stuck up with one or two ...
0
votes
1answer
364 views

Give a combinatorial proof that $\sum_{k=1}^{n} {{k} {n \choose k}^2 ={ n} {{2n-1} \choose {n-1}}}$ [duplicate]

$$\sum_{k=1}^{n} {{k} {n \choose k}^2 ={ n} {{2n-1} \choose {n-1}}}$$ How would I approach this problem to make a combinatorial proof?
2
votes
2answers
51 views

Prove that using induction that $\binom22+\dots+\binom n2 = \binom{n+1}2$ [duplicate]

so I have this math problem where I have to prove this using induction. ...
3
votes
2answers
55 views

What's the property of this series? Is it special? Coefficients of $\left(x\frac{d}{dx}\right)^n f(x) $

I am think about this expression : $e^{\lambda x \frac{d}{dx}}f(x)$. Let us look at each term in the expansion of the exponential operator $e^{\lambda x \frac{d}{dx}}$, $$\left(x\frac{d}{dx}\right)^n ...
0
votes
1answer
38 views

Number of Lattice paths through some point

I have a problem about lattice paths. Here, I mean we can only use (1,0) or (0,1) as steps. We know the number of lattice paths on an $n\times n$ grid that go through $(i,j)$ is equal to ...
1
vote
1answer
60 views

Fun Proof! Show that there are ${m+n \choose n}$ allowable paths from $(0,0)$ to $(m,n)$ for all $m, n \in Z$

Define an ``allowable path" from a point $(x,y) \in R^2$ to a point $(x',y') \in R^2$ to be a path from $(x,y)$ to $(x',y')$ consisting of a finite sequence of positive, length $1$, horizontal and ...
0
votes
1answer
75 views

Is there a closed formula for $\binom{a}{k}+\binom{b}{k}-\binom{c}{k}$?

For integers $c > b > a > k \ge 1$, consider the binomial sum $$\binom{a}{k}+\binom{b}{k}-\binom{c}{k}. \tag{$\star$}$$ Does ($\star$) have other closed-form representations?
2
votes
2answers
36 views

Proof of Identity Involving Binomial Coefficients

I am new to stack exchange. I can't find a duplicate of this problem (some similar but I am stuck at a specific place!). I need to prove: $\binom{n}{r} = \frac{n-r+1}{r} \binom{n}{r-1}$ I know that ...
3
votes
5answers
177 views

What is the sum of the series with binomial sequences: $\sum_{k=0}^{n} k \binom{n}{k}$? [duplicate]

compute this sum: $\sum_{k=0}^{n} k \binom{n}{k}$ I tried but I got stuck
1
vote
2answers
360 views

Stirling's Approximation for binomial coefficient

In this proof, it is assumed that, for $k << n$, ${n \choose k} \approx \frac{n^k}{k!}$, given Stirling's approximation. How does Stirling's Approximation, in either form $\ln n! \approx ...
2
votes
2answers
51 views

Intuitive explanation of binomial coefficient formula

Regarding the formula for binomial coefficients: $\binom{n}{k}=\frac{n(n-1)(n-2)...(n-k+1)}{k!}$ the professor described the formula as first choosing the k objects from a group of n, where order ...