Combinatorics problem help.........

The total number of ways in which 5 balls of different colors can be distributed among 3 persons so that each person gets at least one ball is

(A) 75 (B) 150 (C) 210 (D) 243

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

First consider how you can distribute $5$ balls over $3$ players while every player gets at least $1$ ball, where all balls are equal and all players are equal. You could give one player $3$ balls and the other players both $1$ ball, or you could give one player $1$ ball and the other players both $2$ balls. If you work out these cases separately, you'll find the answer.

Tim Vermeulen - 7 years, 10 months ago

answer is 150. in how many ways can you distribute 5 balls among 3 people such that atleast one of them gets 1? it is nothing but 3,1,1 and 2,2,1. so the desired answer will be (5!/3!)(3!/2!)+ (5!/ (2!)^2)(3!/2!) you need to multiply by 3!/2! because there are these many ways to arrange 3,1,1 or 2,2,1 . By evaluating the expression answer will come as 150.

subharthi chowdhuri - 7 years, 10 months ago

3^5 =243. (D)is the answer.

Bhargav Das - 7 years, 11 months ago

how to do it????

Rajath Krishna R - 7 years, 11 months ago

Ball 1 could go to person A, or B, or C -> 3 choices

Ball 2 could go to person A, or B, or C -> 3 choices

...

Ball 5 could go to person A, or B, or C -> 3 choices

$3*3*3*3*3 = 3^{5} = 243$

Luca Bernardelli - 7 years, 10 months ago

@Luca Bernardelli – Note that every player gets at least one ball.

Tim Vermeulen - 7 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$