Question

A mathmatics journal has accepted 14 articles for publication.however ,due to budgetary restraints only 9 articles can be published this month.how many ways can the journal editor assemble 9 of the 14 articles for publication?

Answer 1

Questions like this are answered by the binomial coefficient. The binomial coefficient $\binom{n}{k}$ (read "n choose k") means exactly what you're looking for: "How many ways are there to choose k elements from a set of n elements?"

In other words, it counts the number of possible subsets of cardinality k from a set of cardinality n. It is defined as follows:

$\displaystyle \binom{n}{k} = \dfrac{n!}{k!(n-k)!}$

Where n! is the factorial of n:

$n! = n(n-1)(n-2)(n-3)\ldots 4 \cdot 3 \cdot 2$

So, you have

$\displaystyle \binom{14}{9} = \dfrac{14!}{9!5!} = \dfrac{14\cdot13\cdot12\cdot11\cdot10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2}{9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot5\cdot4\cdot3\cdot2}$

Of course, a lot of factors simplify:

$\dfrac{14!}{9!5!} = \dfrac{14\cdot13\cdot12\cdot11\cdot10}{5\cdot4\cdot3\cdot2} = 14\cdot13\cdot11 = 2002$