Mind Riddles: combinations

TigerLilly

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Oct 16, 2005
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I wanted to see if I was correct with these three questions:

1) You have 5 identical boxes that stand one after another. How many different ways can you put your 3 identical eggs into these boxes?

I found a few different answers for this one. You have 3 identical eggs and 4 possible ways of putting 3 eggs into a box (0, 1, 2, or 3). And with 5 identical boxes, would this mean a total of 100 different ways?

2) How many ways can the letters of the word "YUSSUF" be arranged?

For this one, I got an answer of 30 different ways. You 6 total letters and a full total of 720 different rearrangements. Then, you have two letters that are the same so 4 different letters and 24 different ways. Then in dividing the total number of possible rearrangements by 24 real combinations, you'd get an answer of 30.

3) A soccer team includes 18 players. On the field, there should be only 11 players for that team. How many different groups of 11 players can the coach choose?

I had no idea on that last one...
 
I wanted to see if I was correct with these three questions:

1) You have 5 identical boxes that stand one after another. How many different ways can you put your 3 identical eggs into these boxes?

I found a few different answers for this one. You have 3 identical eggs and 4 possible ways of putting 3 eggs into a box (0, 1, 2, or 3). And with 5 identical boxes, would this mean a total of 100 different ways?

2) How many ways can the letters of the word "YUSSUF" be arranged?

For this one, I got an answer of 30 different ways. You 6 total letters and a full total of 720 different rearrangements. Then, you have two letters that are the same so 4 different letters and 24 different ways. Then in dividing the total number of possible rearrangements by 24 real combinations, you'd get an answer of 30.

3) A soccer team includes 18 players. On the field, there should be only 11 players for that team. How many different groups of 11 players can the coach choose?

I had no idea on that last one...

For the second one, you said there were four of the same letters, actually what you have to do is

6!/2!2! since there are two "U"'s and two "S"'s. so thats two of the smae letters (2!) times two of the same letters(2!)

and so you should do 720/4

and for the last one, you would use a thing called a permutation, which means you would do n!/(n-p)! n=things to choose from p=positions to fill

so in this case you have n=18 players and p=11 positions so

18!/(18-11)! and now you can do the rest!
 
Hello, TigerLilly!


1) You have 5 identical boxes that stand one after another.
How many different ways can you put your 3 identical eggs into these boxes?

I found a few different answers for this one.
You have 3 identical eggs and 4 possible ways of putting 3 eggs into a box (0, 1, 2, or 3).
And with 5 identical boxes, would this mean a total of 100 different ways.
I don't follow your reasoning.

I assume that "stand one after another" means "in a row".
Then the boxes are not identical.
They are distinguishable: 1st box, 2nd box, 3rd box, 4th box, 5th box.


There are three possible scenarios . . .

[1] All 3 eggs in one box:.\(\displaystyle 5\) ways.

[2] Two in one box, one in another: .\(\displaystyle 5\cdot4 \,=\,20\) ways.

[3] One egg in each of 3 boxes: .\(\displaystyle _5C_3 \,=\,{5\choose3} \,=\,10\) ways.


There are: .\(\displaystyle 5 + 20 + 10 \:=\:35\) ways.
 
For the second one, you said there were four of the same letters, actually what you have to do is 6!/(2!2!)........(You must have grouping symbols here, otherwise, what you have typed is equivalent to 720.)...
...
 
1) You have 5 identical boxes that stand one after another. How many different ways can you put your 3 identical eggs into these boxes?


I have a simple question for all of you: why would a question say identical boxes if that is not what it means?

If that is what it means, then the answer is three: 3; 2, 1; & 1,1,1.

This is a well know problem known as integer-partitions studied extensively, by Ivan Niven among many.
How many ways can an positive integer be divided into distinct summands?

Were problem:
1) You have 5 identical boxes. How many different ways can you put your 6 identical eggs into these boxes?

The answer is 10. There are ten ways to partition six into five or fewer summands.
 
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