Question

MEDAL+FAN+TESTIMONIAL

The calculation 9!/3!6! is equivalent to 9C3 (or 9C6)


Explain clearly why you could solve this question using combinations, and why this is equivalent to considering permutations with repeated items.

Answer 1

@ganeshie8

Answer 2

@nincompoop

Answer 3

Oh thanks!

Answer 4

Thanks for your previous explanations! They were easy to understand:)

Answer 5

What can I do to get an overview of this question?

Answer 6

Hey is that the complete problem ?

Answer 7

We have looked at situations in which we need to determine the number of possible routes between two places. We can look at the situation below as 9 steps, six of which must be East and three of which must be South.


This gives us 9!/3!6! = 84 possible routes

Answer 8

Those are the starting places:)

Answer 9

Thanks for pointing them out

Answer 10

The file "imagen" is exactly as it is

Answer 11

What may be the discernible connection here?

Answer 12

Lets work this problem using "permutations" only first

Answer 13

Ok

Answer 14

maybe we need to calculate the 9!/3!6! first?

Answer 15

6 east and 3 south.

Answer 16

A valid route is 9 steps long.
Must have 6 East and 3 South, yes ?

Answer 17

Yes. They have to be the shortest.

Answer 18

I got 6 east 3 south too

Answer 19

Like :
\[EEEEEE~SSS\]

Answer 20

9P9?

Answer 21

How many distinct words can you make by rearranging the characters in that string above ?

Answer 22

9!/(6!*3!)

Answer 23

Because of the indistinguishable letters

Answer 24

How did you get that, could you please explain a bit more..

Answer 25

wasn't that 9!/(6!)(3!) from the original problem?

Answer 26

Because there are 9 letters to be arranged, but permutation 9! would have to be reduced by letters that are identical to each other. And 6! accounts for Es and 3! accounts for Ss. Therefore the total number of distinct words are divided by overlapping ones.

Answer 27

I had a similar question before where I had to rearrange letters to make distinct words, and one of the math tutors here showed me that I need to divide the original all possible options by the factorials of overlapping ones.

Answer 28

Perfect!

Answer 29

so that is a method using permutations only, lets look at working the total number of routes using "combinations" only

Answer 30

For example, in "mathematics"
there are 2 "m"s
2 identical "a"s
2 identical "t"s

So
11!/(2!*2!*2!)=possible rearranged distinct words from mathematics.

Answer 31

Ok.

Answer 32

\[\frac{n!}{r!(n!-r!)}\] is the formula for the number of possible combinations of r objects in from the set of n objects

it's usually written in nCr.

Answer 33

To go from red point on top left corner to the green point on bottom right corner, you need to take 9 steps. 3 of them South and 6 toward East. So if you `choose 3 steps for South out of total 9 steps`, the remaining 6 steps are forced to be East.

Answer 34

3 steps south where it could be anywhere, but 6 steps east would have to be made to get to the destination