Questions tagged [multinomial-theorem]
An extension to the binomial theorem. It gives the expansion of a multinomial $(x_0,\dots,x_{m-1})^n$.
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Finding number of dissimilar terms in an expansion
The question is
The number of dissimilar terms in the expansion of $(1+x^3+x^4)^4$ is ?
Upon using the formula $^{n+r-1}C_{r-1}$ or using permutation & combination, I am getting 15 as the answer ...
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Number of integral solutions for $x_1 + x_2 - x_3 = n$ where $n \geq x_1 , x_2 , x_3 \geq 0$
I have been asked
Integral solutions for $x_1 + x_2 - x_3 = n$ where $n \geq x_1 , x_2 , x_3 \geq 0$.
My approach:
We have, $0 \leq x_3\leq n$
$\Rightarrow n \leq x_1 + x_2 \leq 2n$
...
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Number of monotonically increasing functions such that $f(i)\le i$.
Problem: Consider $n \in \mathbb{N}^+$, set $A = \mathbb{N}^+ _{\leq n}$. Find the number of monotonically increasing functions $f: A → A $ such that $f(i) \leq i$.
I tried using the multinomial ...
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Showing the quintessential logarithm property using the Maclaurin series of $\log$
For $-1\le x<1$, we have
$$\log(1-x) = -\sum_{k=1}^{\infty} \frac{x^{k}}{k}\\$$
Taking $a,b$ with $|a|,|b|<1$ and $(1-a)(1-b)\le2$, on the function side clearly we have
$$\log(1-a)+\log(1-b) = \...
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Why do Bell Polynomial coefficients show up here?
The multinomial theorem allows us to expand expressions of the form ${\left( {{x_1} + {x_2} + {x_3} + {x_4} + ...} \right)^n}$. I am interested in the coefficients when expanding ${\left( {\sum\...
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If $(1+px+x^2)^n=1+a_{1}x+a_{2} x^2+....+a_{2n}x^{2n}$, then prove that $(np-pr)a_{r}=(r+1)a_{r+1}+(r-1-2n)a_{r-1}$ for $1<r<2n$
If $(1+px+x^2)^n=1+a_{1}x+a_{2} x^2+....+a_{2n}x^{2n}$, then prove that $(np-pr)a_{r}=(r+1)a_{r+1}+(r-1-2n)a_{r-1}$ for $1<r<2n$
My try:
I tried putting $r=2$ and solved the problem and verfied ...
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$(1+2x+3x^2)^{50}=a_{0}+ a_{1}x+ ... +a_{100}x^{100}$ Find ratio between $a_{51}$ and $a_{49}$.
$$(1+2x+3x^2)^{50}=a_{0}+ a_{1}x +...+ a_{100}x^{100}$$
Find ratio between $a_{51}$ and $a_{49}$.
Another sub question asked is to find the relation between
$a_{n}, a_{n-1}, a_{n-2}$
My approach
I ...
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Multinomial Theorem expansion in Combinations Problem
While studying application of Multinomial theorem in PnC I got stuck in two questions :
In how many ways the sum of upper faces of four distinct dice can be six ?
The textbook gave the following ...
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Proving the Number Theory Property: $\delta(n^k) = \delta(\delta(n)^k)$ for natural $n$ and $k$
I'm working on a number theory problem for the Regional Mathematical Olympiad, Stage 2 of the Indian Olympiad Programme, and it's been challenging to solve.
The problem is as follows.
A function $\...
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Central Binomial Coefficients and Multinomial Coefficients
Premise
I was looking at the multinomial coefficients when selecting by a specific rule. Then analyzing the sum.
Given the multinonial theorem ($n > 0$):
$$
(x_1+\ldots+x_n)^n = \sum_{k_1+\ldots+...
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Multinomial theorem for a power series
I was wondering if there is a version of the multinomial theorem for the expression:
$$
(1+\sum_{k=1}^\infty a_k x^k)^n.
$$
Thanks in advance.
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The proof by induction of the multinomial theorem
I looked at the proof by induction of the multinomial theorem on Wikipedia and do not understand how to get the last step. Specifically, I do not know why this equality is true:
$$\sum_{k_1 + k_2 + \...
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Five different games are to be distributed among $4$ children randomly. The probability that each child get at least one game is?
Here's a question :-
Five different games are to be distributed among 4 children randomly. The probability that each child get at least one game is?
To find the answer, we will have to find the sample ...
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In the multinomial expansion of $(a+b+c+d)^8$, how many terms (monomials) have coefficient $\begin{pmatrix} 8 \\ 2,4,0,2\end{pmatrix}$?
Can someone please explain why the ansewr is 12?
My current working:
Using the multinomial theorem, each term in the expanded (unsimplified) form would be uniquely determined by its distinct ordered ...
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When not to use $\binom{n + r − 1} {r− 1}$?
The number of non-negative integral solutions of the equation
$x_1+x_2+x_3+....+x_r = n$
is $\binom{n + r − 1} {r− 1}$
I tried using it in the following two questions.
Let $n_1<n_2<n_3<n_4&...
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