se7en14 wrote:
how many ways can a committee of 3 be selected from 7 so that there is a president, a vice president, and a secretary?
Answer is: 210
7x6x5=210
7P3 = 7X6X5 =7!/4!
Why is it 7!/4! ? I don't understand the 4! part.
Can someone explain that for me"?
Thanks!
Out of 7, 3 people need to be selected and arranged (into 3 different positions: President, VP, Sec)
So there are 2 ways to do it: 7 * 6* 5 (you say, select the president in 7 ways, now select VP in 6 ways and then select Sec in 5 ways) = 210
or you can use the permutation formula nPr such that nPr = n!/(n-r)!. nPr helps you select r people out of n people AND arrange those r people.
Above, we used 7P3 = 7!/(7 - 3)! = 7!/4! = 7*6*5 = 210
Now I am assuming that your question is why the formula is n!/(n-r)!
Say you have n people and you want to arrange them. You can do it in n! ways, right? Just our basic counting principle. Say there at 7 people and you want to arrange all 7 in 7 spots. You can do it in 7! ways ( = 7*6*5*4*3*2*1). 7 ways to fill the first spot, 6 ways to fill the second. 5 ways to fill the third, 4 ways to fill the fourth etc.
Now what if you have only 3 spots? You have to fill 3 only. You can do it in 7*6*5 ways. What about the rest of the 7-3 = 4 spots? (which is n - r) You have to ignore them. So if you do arrange people in 7 spots by using 7! in the numerator, you must divide out the extra n - r spots i.e. 4!. That is the reason you divide by 4! here.