Find the number of different ways that an instructor can choose 7 students from a class of 35 students for a field trip.
Find the number of different ways that an instructor can choose 7 students from a class of 21 students for a field trip.
Well, if I were the instructor, I would first ask if all 35 students want to go on the field trip. Because if not, I wouldn't want to force anyone to come along against their will.
But let's assume all 35 students are willing participants. In that case, the number of different ways the instructor can choose 7 students can be calculated using the combination formula.
So, the number of ways can be expressed as 35 C 7, which equals a pretty large number! I don't think I can even count that high. How about we just agree that there are a LOT of possibilities and leave it at that?
To find the number of different ways the instructor can choose 7 students from a class of 35 students for a field trip, we can use the concept of combinations.
The number of combinations, denoted as C(n, r), represents the number of ways to choose r items from a set of n items, without regard to their order. In this case, we want to choose 7 students from a class of 35 students, so we need to calculate C(35, 7).
The formula for calculating combinations is given by:
C(n, r) = n! / (r!(n - r)!)
Where n! (read as "n factorial") represents the product of all positive integers less than or equal to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
Now, let's substitute the values into the formula:
C(35, 7) = 35! / (7!(35 - 7)!)
Calculating each factorial separately:
35! = 35 x 34 x 33 x ... x 2 x 1
7! = 7 x 6 x 5 x ... x 2 x 1
(35 - 7)! = 28! = 28 x 27 x 26 x ... x 2 x 1
Simplifying further:
C(35, 7) = (35 x 34 x 33 x ... x 2 x 1) / ((7 x 6 x 5 x ... x 2 x 1) x (28 x 27 x 26 x ... x 2 x 1))
Now we can plug in the values and calculate using a calculator or spreadsheet software:
C(35, 7) = (1,033,896,200) / (5,040 x 3,265,920)
Simplifying this fraction will give us the final answer.
Find the number of different ways that an instructor can choose 4 students from a class of 19 students for a field trip. Hint: The order of selection of the students for the trip does not matter.
The word "choose" is usually a good indicator that your are dealing with combinations rather than permutations, that is, the order in which the students are selected does not matter, so
35 choose 7 or C(35,7) or 35!/(7!28!) or 6,724,520