Student examination
How many ways can a teacher select a group of 6 students to sit in the front row if the class has 13 students?
Correct answer:
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Someone
the question doesnt say that the ordering doesnt matter, so wouldnt that mean the answer would be 13P6 or 1235520 ways?
1 year ago 2 Likes
Dr. Math
To determine the number of ways a teacher can select a group of 6 students from a class of 13 students, we will use the concept of combinations. Combinations are used when the order of selection does not matter.
We need to select 6 students out of 13, and the order in which they are selected does not matter. This is a combination problem.
The number of ways to choose k items from n items is given by the combination formula:
where:
- n! (n factorial) is the product of all positive integers up to n ,
- k! is the factorial of k ,
- (n - k)! is the factorial of (n - k) .
Here, n = 13 (total students) and k = 6 (students to select). Substitute these values into the formula:
We can simplify the factorials by canceling out common terms:
The 7! terms cancel out:
Now, calculate 6! :
Calculate the numerator:
So, the numerator is 1,235,520.
Now, divide the numerator by the denominator:
The number of ways to select a group of 6 students from 13 is:
Step 1:
Understand the ProblemWe need to select 6 students out of 13, and the order in which they are selected does not matter. This is a combination problem.
Step 2:
Recall the Combination FormulaThe number of ways to choose k items from n items is given by the combination formula:
C(n, k) = n!k!(n - k)!,
where:
- n! (n factorial) is the product of all positive integers up to n ,
- k! is the factorial of k ,
- (n - k)! is the factorial of (n - k) .
Step 3:
Substitute the ValuesHere, n = 13 (total students) and k = 6 (students to select). Substitute these values into the formula:
C(13, 6) = 13!6!(13 - 6)! = 13!6! · 7!.
Step 4:
Simplify the FactorialsWe can simplify the factorials by canceling out common terms:
C(13, 6) = 13 × 12 × 11 × 10 × 9 × 8 × 7!6! × 7!.
The 7! terms cancel out:
C(13, 6) = 13 × 12 × 11 × 10 × 9 × 86!.
Now, calculate 6! :
6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.
Step 5:
Calculate the NumeratorCalculate the numerator:
13 × 12 = 156,
156 × 11 = 1716,
1716 × 10 = 17160,
17160 × 9 = 154440,
154440 × 8 = 1235520.
So, the numerator is 1,235,520.
Step 6:
Divide by the DenominatorNow, divide the numerator by the denominator:
C(13, 6) = 1,235,520720 = 1,716.
Final Answer:
The number of ways to select a group of 6 students from 13 is:
1,716
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