A mathematics journal has accepted 14 articles for publication. However, due to budget restraints, only 7 articles can be published this month. How many specific ways can the journal editor assemble 7 of the 14 articles for publication?
Possible answers:
14
3,432
98
17,297,280
To find the number of specific ways the journal editor can assemble 7 of the 14 articles for publication, we can use the combination formula. The formula for calculating combinations is:
nCr = n! / (r!(n-r)!)
Where n is the total number of items available (14, in this case) and r is the number of items to be selected (7, in this case).
So, we can calculate the number of ways as:
14C7 = 14! / (7!(14-7)!)
= 14! / (7!7!)
Now, we can simplify the expression:
14! = 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7!
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1
Substituting these values into the formula:
14C7 = (14 x 13 x 12 x 11 x 10 x 9 x 8 x 7!) / (7 x 6 x 5 x 4 x 3 x 2 x 1 x 7!)
The 7! terms in the numerator and denominator cancels out:
14C7 = (14 x 13 x 12 x 11 x 10 x 9 x 8) / (7 x 6 x 5 x 4 x 3 x 2 x 1)
Now, let's calculate this expression:
14C7 = (14 x 13 x 12 x 11 x 10 x 9 x 8) / (7 x 6 x 5 x 4 x 3 x 2 x 1)
= 17,297,280
Therefore, the correct answer is 17,297,280.
To find the number of ways the journal editor can assemble 7 articles out of the 14 accepted articles, we can use the concept of combinations.
The number of ways to choose a subset of size k from a set of n elements is given by the binomial coefficient, also known as n choose k, denoted as C(n, k) or nCk.
The formula for the binomial coefficient is: C(n, k) = n! / (k! * (n-k)!)
In this case, we want to choose 7 articles out of the 14 accepted articles, so we can calculate it as: C(14, 7) = 14! / (7! * (14-7)!)
Simplifying the equation:
14! = 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7!
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1
Now, plugging these values into the formula we get:
C(14, 7) = (14 * 13 * 12 * 11 * 10 * 9 * 8 * 7!) / (7 * 6 * 5 * 4 * 3 * 2 * 1 * 7!)
Simplifying further:
C(14, 7) = (14 * 13 * 12 * 11 * 10 * 9 * 8) / (7 * 6 * 5 * 4 * 3 * 2)
Doing the calculation gives us:
C(14, 7) = 17,297,280
Therefore, the journal editor can assemble the articles in 17,297,280 specific ways.
So, the correct answer is 17,297,280.
14 choose 7
=14C7=14!/(7!7!)