# An instructor will choose 3 problems from a set of 7 containing 3 hard and 4 easy problems. If the selection is made at random, what is the probability that only the hard problems are chosen?

11 years ago

11 months ago

## To calculate the probability of only the hard problems being chosen, we need to determine the total number of possible selections and the number of selections where only the hard problems are chosen.

Step 1: Determine the total number of possible selections:

Since the instructor chooses 3 problems from a set of 7, we can use the combination formula to find the total number of possible selections. In this case, we want to select 3 out of the 7 problems, regardless of the difficulty level.

The combination formula is given by: C(n, r) = n! / (r!(n-r)!)

where n is the total number of items to choose from, and r is the number of items to be chosen.

Plugging in the values, we have:

C(7, 3) = 7! / (3!(7-3)!)

= 7! / (3!4!)

= (7 x 6 x 5 x 4!) / (3 x 2 x 1 x 4!)

= (7 x 6 x 5) / (3 x 2 x 1)

= 35

So there are a total of 35 possible selections.

Step 2: Determine the number of selections where only the hard problems are chosen:

Since there are 3 hard problems in a set of 7, the instructor needs to choose all 3 hard problems and no easy problems.

The number of ways to choose 3 hard problems is simply 1 (because there is only one set of 3 hard problems).

The number of ways to choose 0 easy problems (since the instructor shouldn't choose any easy problems) can be calculated using the combination formula as well:

C(4, 0) = 4! / (0!(4-0)!)

= 4! / (0!4!)

= (4 x 3 x 2 x 1) / (1 x 1 x 1)

= 4

So there is only 1 way to choose all 3 hard problems and no easy problems.

Step 3: Calculate the probability:

The probability of the desired event (only the hard problems being chosen) is given by:

Probability = (Number of successful outcomes) / (Total number of possible outcomes)

Number of successful outcomes = 1 (since there is only 1 way to choose all 3 hard problems and no easy problems)

Total number of possible outcomes = 35

Therefore, the probability is:

Probability = 1 / 35

Hence, the probability that only the hard problems are chosen is 1/35.