All Questions
23
questions
19
votes
5
answers
1k
views
Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity:
$$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$
for all natural ...
- 809
6
votes
3
answers
750
views
Preventing "proof by homework"?
I am doing problem 3d in the Prologue of Spivak:
Prove $(a+b)^n = a^n + {n\choose1}a^{n-1}b + {n\choose2}a^{n-2}b^2 + ... + {n\choose n-1}ab^{n-1} + b^n$
I feel like my proof could pass off as ...
- 3,076
6
votes
2
answers
144
views
How to show $\sum\limits_{r=0}^n \frac{1}{r!} \lt\left (1 + \frac{1}{n}\right)^{n+1}$ for all $n \ge 1$?
Using the binomial expansion, it is quite is easy to show that $$\left(1+\frac{1}{n}\right)^n \le \sum_{r=0}^{n} \frac{1}{r!} $$ for all $n\in\mathbb{Z^+}$, with equality holds when $n=1.$ (Can it be ...
3
votes
4
answers
163
views
Proving $\sum_{i=0}^n (-1)^i\binom{n}{i}\binom{m+i}{m}=(-1)^n\binom{m}{m-n}$
I am trying to prove the following binomial identity:
$$\sum_{i=0}^n (-1)^i\binom{n}{i}\binom{m+i}{m}=(-1)^n\binom{m}{m-n}$$
My idea was to use the identity
$$\binom{m}{m-n}=\binom{m}{n}=\sum_{i=0}^n(-...
- 87
3
votes
2
answers
315
views
Solving binomial summation $\sum_{k=0}^{\lfloor{n/2}\rfloor} \binom{n-k}{k} 2^{n-k}$
How can we solve the sum
$$\sum_{k=0}^{\lfloor{n/2}\rfloor} \binom{n-k}{k} 2^{n-k}$$
The problem arose from a counting question, but I am unable to solve this sum.
Edit:
The counting problem was ...
- 1,334
3
votes
1
answer
139
views
Double summation with binomial coefficients
Question :
Find the value of the following expression :
$$ \frac{\sum_{i=0}^{2024}\sum_{r=0}^{2024}(-1)^r{2024 \choose r}(2024-r)^i}
{\sum_{r=0}^{2025}(-1)^r\binom{2025}{r}(2025-r)^{2025}} $$
I am not ...
- 37
3
votes
2
answers
117
views
$1\binom{20}1+2\binom{20}2+3\binom{20}3+\dots+19\binom{20}{19}+20\binom{20}{20}$
$$1\binom{20}1+2\binom{20}2+3\binom{20}3+\dots+19\binom{20}{19}+20\binom{20}{20}$$
I solved it by letting the sum be $S$, then adding the sum to itself but taking the terms from last to first and then ...
2
votes
2
answers
1k
views
Use induction and Newton's binomial formula to show that $\binom{n}{0}+\binom{n}{1}+\cdot+\binom{n}{n}=2^n, \forall n\in \mathbb N$ [duplicate]
Use induction and Newton's binomial formula to show that:
$ i)$ $ \binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{n}=2^n, \forall n\in \mathbb N$
$ ii)$ $\binom{n}{0}-\binom{n}{1}+\binom{n}{3}-\cdots+(-1)...
- 1,317
2
votes
2
answers
46
views
How to write $\left(\frac{A+Bs}{C+Ds}\right)^N$ as $\sum_{n=-\infty}^{\infty} s^n P_n$?
We are given this thing
$$J=\left(\frac{A+Bs}{C+Ds}\right)^N$$
Where $A,B,C,$ and $D$ are non-zero constants, $N$ is a positive constant.
We are told to find $P_0$ (the coefficient of the term $s^0$...
- 1,519
2
votes
1
answer
59
views
Roots of a summation
I'm trying to solve an equation:
$$\sum_{n=0}^b \left(\left(\frac{a+xn}{b}\right)\binom{b}{n}(-x)^{b-n}\right)=0$$
Where a and b are constants.
I thought of solving it by using the binomial theorem. ...
- 119
2
votes
1
answer
111
views
Need help with inductive proof of Binomial Theorem
I'm new to math and trying to learn about the Binomial Theorem, by following this tutorial.
I got stuck trying to read the Induction Proof.
They give an example of using the Sum notation:
$$ (x + y)^...
- 23
2
votes
2
answers
70
views
Question on changing the index of summation
$$b(a+b)^m = \sum_{j=0}^m \binom{m}{j}a^{m-j}b^{j+1}= \sum_{k=1}^m \binom{m}{k-1}a^{m+1-k}b^{k}+b^{m+1}$$
I believe $j = k-1$ though the book does say that.
This is related to proving the binomial ...
- 21
1
vote
3
answers
73
views
Writing $(m+2)^n-(m-2)^n$ in summation notation.
I have expanded $(m+2)^n-(m-2)^n$ the following way:
$$(m+2)^n-(m-2)^n = 2 {n \choose 1}m^{n-1}+ \dots + {n \choose n-1}m2^{n-1}-{n \choose n-1}m(-2)^{n-1}+{n \choose n}2^n - {n \choose n}(-2)^n$$
...
- 221
1
vote
1
answer
45
views
Can this be done without substitution of values?
If $${ \left( 1-{ x }^{ 3 } \right) }^{ n }=\sum _{ r=0 }^{ n }{ { a }_{ r }{ x }^{ r }{ \left( 1-x \right) }^{ 3n-2r } } $$ then find $a_r$.
My first attempt:
I wrote the above equation as:
$$\...
- 1,511
1
vote
2
answers
150
views
Solve following summation using Binomial theorem
I apologise in advance for posting another one of these homework assignment-questions; this one is pissing me off.
The question is: solve $\sum_{k=0}^n {{n \choose k}} k 4^k $ using the binomial ...
