Rosen chapter 5.5 question 50

Question

How many ways are there to distribute 5 distinguishable objects into three indistinguishable boxes?

Comments

pls ans

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21 ways.

$\left ( 5,0,0 \right )=3 ways$

$\left ( 4,1,0 \right )=3! ways=6 ways$

$\left ( 3,2,0 \right )=3! ways=6 ways$

$\left ( 2,2,1 \right )=3 ways$

$\left ( 1,3,1 \right )=3 ways$

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41 ways.

(5,0,0) = 1 way

(4,1,0) = 5 ways

(3,2,0) = 10 ways

(2,2,1) = 15 ways

(1,3,1) = 10 ways

Answer 1

refer : https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind

this can be done by Stirling number formula of combinatorics

Answer 2

Using the usual method of bars and stars, we can arrange 5 objects and 2 walls of containers in 7! ways. Now, 2 walls are same.Also, the boxes are indistinguishable.

So, ans = $\frac{7!}{2! * 3!}$

Answer 3

See.
https://math.stackexchange.com/questions/340890/how-many-ways-are-there-to-distribute-5-balls-into-3-boxes-under-additional-con/340957