Question

In a state lottery, if 6 numbers are chosen out of 50. If you match all 6, you win a million dollars. Is this a combination or permutation? How do you know? If it was the other one, would your chances be higher or lower and as a result would the pay out be higher or lower?

Answer 1

Since whether the order of the 6 numbers chosen is important or not is not mentioned so we would assume that the order is not important and hence, this is a combination.

The total number of ways in which 6 numbers can be chosen out of 50 is given by nCr

If the order of the 6 numbers chosen was important, then it would be permutation and not just combination. In this case, after choosing the 6 numbers, they can also be permuted among themselves to give 6! permutations of the same 6 numbers which would all be considered different.

The total number of ways in which 6 numbers can be chosen and permuted among themselves is given by nPr.

Since the number of possibilities in the case of permutation is very much higher than that of combination, hence in case of permutation, the chances of winning will be lower.

The expected payout is given by multiplying the winning amount with the probability of winning.

The expected payout in case of the combination will be

= $0.063

The expected payout in case of permutation will be

= $0.0000875

Hence, the payout will be lower in the case of permutation.