# Solved: Proving Distributions of Musicians w/ Induction

• rsa58
In summary, the conversation discusses the use of induction to prove a statement about distributing musicians to orchestras. The original question states that there are k musicians to be distributed among n orchestras, with exactly ki musicians in the ith orchestra. The goal is to prove that there are k!/(k1! · · · kn!) different distributions possible. The conversation mentions using the choose function to find all possible distributions, but the professor suggests using induction by selecting from a pool of k and checking for n=1 and n=2. The person asking for help is unsure how to proceed with this method and is only able to find a proof using the choose function.
rsa58
[SOLVED] induction? question

## Homework Statement

the question is:
let k, n, and k1, . . . , kn be given natural numbers, such that
k1 + . . . + kn = k.
Assume that k musicians shall be distributed to n orchestras such that exactly
ki musicians play in the ith orchestra. Prove that there exist exactly
k!/(k1! · · · kn!)
different distributions.

is it possible to use induction to answer this? i can prove it by using the choose function to find all the possible distributions. in that way i get a proof for the statement, but i am unable to assume that it is correct for n, and then show it is correct for n+1. can someone give me some ideas?

## The Attempt at a Solution

in that way i get a proof for the statement
Why do you need induction if you have a proof without it?

because the professor said something about induction. he said to "select from a pool" (of k's i guess, whatever that means) and then use induction over n. he said to check for n=1 and n=2 and show that the statement holds for n>= 2. he does seem kind of out of it though. to use induction how would i proceed? i need to combine some expression with
k!/(k1! · · · kn!)

right? i can't find this expression. the only sort of induction i can get is a logically inherent one using the choose function.

## What is "Solved: Proving Distributions of Musicians w/ Induction"?

"Solved: Proving Distributions of Musicians w/ Induction" is a scientific study or research project that aims to prove the distribution of musicians based on certain characteristics or variables using the method of mathematical induction.

## Why is proving distributions of musicians important?

Proving distributions of musicians can provide valuable insights into the demographics and trends of the music industry. It can also help in identifying potential biases or inequalities in the representation of different groups of musicians.

## What is mathematical induction and how is it used in this study?

Mathematical induction is a proof technique used to prove a statement or proposition for all natural numbers. In this study, mathematical induction is used to prove the distribution of musicians by analyzing the patterns and trends observed in a specific group of musicians and extending it to the entire population.

## What are the potential limitations of using mathematical induction in this study?

One potential limitation of using mathematical induction is that it relies on the assumption that patterns observed in a specific group of musicians will also be observed in the entire population. This may not always be the case, leading to inaccurate conclusions. Additionally, the accuracy of the results also depends on the quality and reliability of the data used in the study.

## How can the results of this study be applied in the real world?

The results of this study can be applied in various ways, such as informing diversity and inclusion initiatives in the music industry, identifying potential areas for growth or improvement, and providing insights for marketing and targeting specific demographics in the music market.

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