A school dance committee of four people is selected at random from a group of 9 ninth graders, 8 eighth graders, and 10 seventh graders. What is the probability that the committee has all seventh graders? The probability that the committee has all seventh graders is?

NOTE: I realize that this is a combination problem with 10C4 as the set up. But my question is how would I go about multiplying and dividing this out into a fraction? I am a new learner to permutations and combinations.

how many 4 person committees can be selected from the 27 students

... 27C4

how many 4 person committees can be selected from the 10 7th graders
... 10C4

probability of all 7th graders ... 10C4 / 27C4

Well, let's calculate the probability using combinations! To find the probability that the committee has all seventh graders, we need to find the number of favorable outcomes (all seventh graders) divided by the total number of possible outcomes.

The total number of possible outcomes is the number of ways to select a committee of four members from the entire group (9 ninth graders, 8 eighth graders, and 10 seventh graders). This can be calculated using combinations:

Total possible outcomes = total number of ways to select 4 people from a group of 27 (9+8+10)

Now, for favorable outcomes, we want to select only seventh graders. Since there are 10 seventh graders, the number of favorable outcomes is the number of ways to select 4 people from a group of 10 (all seventh graders). This can also be calculated using combinations:

Favorable outcomes = number of ways to select 4 people from a group of 10 (seventh graders)

Now, to find the probability, we divide the favorable outcomes by the total possible outcomes:

Probability = Favorable outcomes / Total possible outcomes

So if we calculate the combinations and divide them, we get:

Probability = (number of ways to select 4 people from a group of 10) / (number of ways to select 4 people from a group of 27)

I hope that answers your question! And remember, even if the probability is low, never lose hope. It only takes one lucky seventh grader to make it happen! Good luck with the committee selection!

To find the probability of selecting a committee with all seventh graders, we need to determine the total number of possible committees that can be formed and the number of committees that consist only of seventh graders.

Total number of possible committees = total number of ways to choose 4 people from the group of 9 ninth graders, 8 eighth graders, and 10 seventh graders.

To calculate this, we use the combination formula:

Total number of possible committees = C(27, 4)

= 27! / (4! * (27-4)!)

= 27! / (4! * 23!)

= (27 * 26 * 25 * 24) / (4 * 3 * 2 * 1)

= 17550

Now, to determine the number of committees that consist only of seventh graders, we need to choose 4 people from the group of 10 seventh graders:

Number of committees with all seventh graders = C(10, 4)

= 10! / (4! * (10-4)!)

= 10! / (4! * 6!)

= (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)

= 210

Finally, to find the probability, we divide the number of committees with all seventh graders by the total number of possible committees:

Probability of selecting a committee with all seventh graders = number of committees with all seventh graders / total number of possible committees

= 210 / 17550

= 0.011965

Therefore, the probability that the committee has all seventh graders is approximately 0.011965 or 1.1965%.

To find the probability that the committee has all seventh graders, we need to first determine the total number of possible committees and then find the number of committees with all seventh graders.

To determine the total number of possible committees, we need to calculate the total number of ways we can select four people from the entire group of ninth, eighth, and seventh graders.

The total number of ways to select four people from a group of (9 + 8 + 10) = 27 students can be calculated using combinations. We can write this as 27C4, which represents choosing 4 students from a group of 27. The formula for combinations is nCr = n! / (r!(n-r)!), where n is the total number of objects to choose from and r is the number of objects to choose.

So, in this case, the total number of ways to select four people from the group of 27 students is:

27C4 = 27! / (4!(27-4)!)
= 27! / (4!23!)

Now, to calculate the number of committees with all seventh graders, we need to find the number of ways to select four people from the group of 10 seventh graders. This can be calculated as 10C4, which represents choosing 4 students from a group of 10.

Therefore, the number of committees with all seventh graders is:

10C4 = 10! / (4!(10-4)!)
= 10! / (4!6!)
= 10 x 9 x 8 x 7 / (4 x 3 x 2 x 1)
= 210

Now, to find the probability of selecting a committee with all seventh graders, we divide the number of committees with all seventh graders by the total number of possible committees:

Probability = Number of committees with all seventh graders / Total number of possible committees

Probability = 210 / 27C4

Now you can substitute the values into the formula and calculate the probability.