Is there any combinatorial intuition to prove this identity $r\cdot \binom{n}{r}=n\cdot\binom{n-1}{r-1}$?
2026-03-29 14:08:06.1774793286
Combinatorial intuition of identity
139 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in COMBINATORICS
- Using only the digits 2,3,9, how many six-digit numbers can be formed which are divisible by 6?
- The function $f(x)=$ ${b^mx^m}\over(1-bx)^{m+1}$ is a generating function of the sequence $\{a_n\}$. Find the coefficient of $x^n$
- Name of Theorem for Coloring of $\{1, \dots, n\}$
- Hard combinatorial identity: $\sum_{l=0}^p(-1)^l\binom{2l}{l}\binom{k}{p-l}\binom{2k+2l-2p}{k+l-p}^{-1}=4^p\binom{k-1}{p}\binom{2k}{k}^{-1}$
- Algebraic step including finite sum and binomial coefficient
- nth letter of lexicographically ordered substrings
- Count of possible money splits
- Covering vector space over finite field by subspaces
- A certain partition of 28
- Counting argument proof or inductive proof of $F_1 {n \choose1}+...+F_n {n \choose n} = F_{2n}$ where $F_i$ are Fibonacci
Related Questions in COMBINATIONS
- Selection of "e" from "e"
- Selection of at least one vowel and one consonant
- Probability of a candidate being selected for a job.
- Proving that no two teams in a tournament win same number of games
- Selecting balls from infinite sample with certain conditions
- Divide objects in groups so that total sum of sizes in a group are balanced across groups
- Value of n from combinatorial equation
- Number of binary sequences with no consecutive ones.
- Count probability of getting rectangle
- Sum of all numbers formed by digits 1,2,3,4 & 5.
Related Questions in COMBINATORIAL-PROOFS
- Proof of (complicated?) summation equality
- Prove combination arguments $c(c(n,2),2) = 3c(n,3)+3c(n,4)$
- A Combinatorial Geometry Problem With A Solution Using Extremal Principle
- What is the least position a club in EPL can finish with 30 wins?
- Find a combinatorial proof for $\binom{n+1}{k} = \binom{n}{k} + \binom{n-1}{k-1} + ... + \binom{n-k}{0}$
- Use combinatorial arguments to prove the following binomial identities
- money changing problem
- $\forall n\in\mathbb N,x>-1,(1+x)^n\ge1+nx$ Using 2nd Derivative
- Combinatorial proof of $\sum\limits_{i=0}^{r} ({m \choose i}) = {{m + r}\choose m}$
- Intersection of $n$ circles and $m$ lines
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Hint: Consider forming a committee from a group of $n$ people where one member of the committee is special, e.g. the chairman of the committee. Try counting this in two different ways.