We have to pick from a selection of objects, and need to know, in how many ways can
objects be picked from a choice of
There are complications. If we are picking in order of preference, then the order of the selection matters. If you are choosing which University to go to, then you will have a first choice, second choice etc, typically up to five choices. There are 116 universities in the UK.
You may pick your first choice from one of 116.
You may pick your second choice from one of 115.
You may pick your third choice from one of 114.
You may pick your fourth choice from one of 113
You may pick your fifth choice from one of 112.
Hence there are 116*115*114*113*112 possible ways of picking five Universities from 116, in order of preference.
The same consideration would apply when considering the number of ways in which contestants can occupy the first three places in a race, since the order obviously matters. In general the number of ways in which a choice ofobjects can be picked from
objects if the order of selection matters is
Suppose that we have to pick from several groups and we want to find the total number of possible choices. A manager of a national football team could be choosing his team to take to the World cup. He has to choose a goalkeeper, defenders, midfielders and strikers. He is allowed to take 23 players, of which 3 must be goalkeepers. Obviously he will have first, second and third choice goalkeeper and the same may apply for each of defender, midfielder and forward positions. He chooses to take 6 defenders, 8 midfielders and 8 strikers.
Suppose he can pick from 10 possible goalkeepers. Then he can pick in
ways with respect to precedence.
Suppose he can pick from 13 possible defenders. Then he can pick in
ways with respect to precedence.
Suppose he can pick from 11 possible midfielders. Then he can pick in
ways with respect to precedence.
Suppose he can pick from 15 possible goalkeepers. Then he can pick in
ways with respect to precedence.
Hence he can choose his squad in
ways altogether.