A random sample of six integers in the interval [1,42] is to be drawn.

The numbers are selected randomly, one after the other, and repeats
are allowed. What is the probability the six numbers in the sample will
all be different?

number of all possible cases with repetition is

42^6

number of cases where they are all different
= 42x41x40x39x38x37

Prob (all different) = (42x41x40x39x38x37)/42^6
= appr .688

To find the probability that the six numbers in the sample will all be different, we first need to determine the total number of possible outcomes and the number of favorable outcomes.

Total number of possible outcomes:
Since the numbers are selected randomly, one after the other, and repeats are allowed, there are 42 possible choices for each of the six numbers in the sample. Therefore, the total number of possible outcomes is 42^6.

Number of favorable outcomes:
To have all six numbers different, the first number can be any of the 42 choices. For the second number, there are 41 remaining choices that are different from the first number. Similarly, for the third number, there are 40 remaining choices, and so on. Therefore, the number of favorable outcomes is 42*41*40*39*38*37.

Now, we can calculate the probability:

Probability = Number of favorable outcomes / Total number of possible outcomes

Probability = (42*41*40*39*38*37) / (42^6)

By evaluating this expression, we can find the probability.