Combination, 4 dices rolled

[vishal.pathak]

How many different combinations of outcomes can you make by rolling four standard (6-sided) dice if the order of the dice does not matter?
OA not available

[shankar.ashwin]

Not quite sure of my answer, anyways.

We have 6 different possibilities here, (Note: Order does not matter)

All 6 different - 1 Possibility
All 6 same - 6 Possibilities.

2 Same - 4 different

Pick 4 different numbers from 6 - 6C4 (and) of the remaining 2 any 1 will be repeated twice, So 2*6C4 = 30

3 Same - 3 Different.

3 different from 6 - 6C3 ways. Of the other 3 any 1 can be repeated 3 times. So, 3*6C3 = 60

4 Same - 2 Different.

2 different from 6 - 6C2 ways. Of the remaining 4, any 1 can be repeated 4 time. So, 4*6C2 = 60

5 Same - 1 different.

1 from 6 numbers. 6C1 ways. And of the remaining 5, any 1 can be repeated 5 times. So, 5*6C1 = 30.

In total we have, 1+6+30+60+60+30 = 187.

[vishal.pathak]

Not quite sure of my answer, anyways.

We have 6 different possibilities here, (Note: Order does not matter)

All 6 different - 1 Possibility
All 6 same - 6 Possibilities.

2 Same - 4 different

Pick 4 different numbers from 6 - 6C4 (and) of the remaining 2 any 1 will be repeated twice, So 2*6C4 = 30

3 Same - 3 Different.

3 different from 6 - 6C3 ways. Of the other 3 any 1 can be repeated 3 times. So, 3*6C3 = 60

4 Same - 2 Different.

2 different from 6 - 6C2 ways. Of the remaining 4, any 1 can be repeated 4 time. So, 4*6C2 = 60

5 Same - 1 different.

1 from 6 numbers. 6C1 ways. And of the remaining 5, any 1 can be repeated 5 times. So, 5*6C1 = 30.

In total we have, 1+6+30+60+60+30 = 187.

Hi Shankar,

I didnt get it. We are rolling 4 dices here. Why are we making cases based on the number of faces on the dice. Shouldn't we make cases based on the number of dice

Regards,
Vishal

[shankar.ashwin]

Oops I read it as 6 dices. Let me try for 4 again

[shankar.ashwin]

Okay same logic as posted above, not sure if I am correct though.

Four slots _ _ _ _ (But order does not matter)

4 Cases:

All 4 have same numbers.

6 cases (1111,2222 and so on)

All 4 have different numbers

We have 6 choices and 4 slots, so 6C4 ways (Combination because arrangement does not matter), So 15 cases.

2 Same 2 Different

Lets first pick 2 different from 6, we have 6C2 = 15 ways.

Now we have 4 numbers remaining, any any 1 of the 4 can be repeated 2 times. So in total, 4*15 = 60 cases.

3 Same - 1 different.

1 number from 6 - 6C1 = 6

Of the remaining 5 numbers, 1 will be repeated 3 times. So we have 5*6 = 30 cases.

Totally, 6+15+60+30 = 111 cases.