Maths(四)---Probability and Statistics(3)

Combinations

An r-combination of elements of a set is an unordered selection of r elements fromthe set. Thus, an r-combination is simply a subset of the set with r elements

例一.
Find the number of combinations of choosing three letters from the five letters A, B,C, D, E.

Solution
Therefore C(5, 3) =10

例二.
How many different ways are there to select two class representatives from a classof 20 students?

Solution
The number of such combinations is C(20, 2) =190.

例三.
In how many ways can a hand of 5 cards be dealt from an ordinary pack of 52playing cards?

Solution
Therefore the number of ways is C(52, 5) = 2,598,960.

例四.
A committee of 5 members is chosen at random from 6 faculty members of themathematics department and 8 faculty members of the computer science department.In how many ways can the committee be chosen if (a) there are no restrictions; (b)there must be more faculty members of the computer science department than thefaculty members of the mathematics department?

Solution
(a)So the number of ways of choosing thecommittee is C(14, 5) = 2002.

(b)
If there are to be more faculty members of the computer science departmentthan the faculty members of the mathematics department, then the followingconditions must be fulfilled.
(i)The number of ways of choosing is C(8, 5) = 56.

(ii)The number of ways of choosing is C(8, 4) × C(6, 1) = 70 × 6 =420.

(iii)The number of ways of choosing is C(8, 3) × C(6, 2) = 56 × 15 = 840
Therefore the total number of ways of choosing the committee is 56 + 420+ 840 = 1316.