Question

A cupboard contains 5 pairs of shoes.4 shoes are drawn one by one at random. Find the probability that their atleast one pair of shoes is drawn?

Answer 1

How many shoes are there in total?

Answer 2

5 pairs of shoes

Answer 3

There are 2 shoes per pair.
So there are 10 shoes in total, right?

Answer 4

yes

Answer 5

Have you learned combinations, yet?

Answer 6

yes i know combination

Answer 7

Then you can easily find the number of ways to choose 4 shoes out of 10 shoes.

Answer 8

it is not directly 4 out of 10 shoes. we need to choose atleast one pair, the answer should be 13/21. can u tell the answer is coming

Answer 9

can u tell how the anwer is coming

Answer 10

We're getting to the answer.
The number of ways to choose 4 shoes out of 10 shoes is in the denominator.

Answer 11

yes what is the numerator

Answer 12

Now, it's saying to find the probability that you pick at least 1 pair of shoe.

Answer 13

If you do 1-(probability that no pair of shoe come out), then that's the same thing as the probability to pick at least 1 pair of shoe.

Answer 14

how to do that

Answer 15

Say that the shoes are a1,a2,b1,b2,c1,c2,d1,d2,e1,e2,f1,f2

Answer 16

ok ok sir

Answer 17

Wait. Sorry.
say that the shoes are a1,a2,b1,b2,c1,c2,d1,d2,e1,e2

Answer 18

we have to remember that the order doesn't matter.

Answer 19

When we pick the first shoe, it can be any of the 10 shoes, right?

Answer 20

ok ok

Answer 21

When we pick the second shoe, we already picked 1 and we can't pick the pair of it, so it has to be from 8 shoes, right?

Answer 22

yes

Answer 23

using similar way, the 3rd time is 6 and the 4th time is 4.

Answer 24

So you might think that there are 10*8*6*4 ways to pick 4 shoes without any pairs.

Answer 25

yes that's right

Answer 26

BUT, I mentioned that order doesn't matter.

Answer 27

For example, (a1,b1,c1,d1) and (b1,a1,c1,d1) are considered the same.

Answer 28

ok

Answer 29

So for every combination we counted 4! times.
So the actual ways to count without picking any pairs can be calculated as (10*8*6*4)/4!

Answer 30

Following me so far?

Answer 31

yes

Answer 32

So, the probability to choose 4 shoes without any pairs is\[\frac{ \frac{ 10*8*6*4 }{ 4! } }{ 10C4 }\].
10C4 is 10 combination 4

Answer 33

Calculate that to get 8/21.
So the probability to pick 4 shoes with at least 1 pair is 1-(8/21)=13/21

Answer 34

i understood everything but why there should be division of 4 factorial. can u explain

Answer 35

Take these 4 combinations
a1,b1,c1,d1
b1,a1,c1,d1
a1,b1,d1,c1
c1,a1,b1,d1

Answer 36

Because the order doesn't matter all 4 of them are same, right?

Answer 37

thank u very much sir

Answer 38

So for that, the number of ways to order those 4 shoes are 4!.
Because we counted 4! combinations as same, we have to divide by 4!

Answer 39

you're welcome :)