Maths(四)---Probability and Statistics(2)

Arrangements
例一.
Find the number of ways of arranging the letters A, B and C.

Solution
3 × 2 × 1 = 3! = 6.
The arrangements are ABC, ACB, BCA, BAC, CAB and CBA.

例二.
It is known that the password on a computer system contain the three letters A, Band C followed by the six digits 1, 2, 3, 4, 5, 6. Find the number of possiblepasswords.

Solution
Therefore the total number of possible passwords is 3! × 6!= 4320, i.e. 4320 different passwords can be formed.


例三.
Find the number of ways of arranging the letters A, B and B.

Solution
Therefore the number of arranging the 3 letters, of which 2 are alike, is3!/ 2!=3

例四.
Find the number of ways that the letters of the word STATISTICS can be arranged.

Solution
Therefore the number of ways is 10!/3!3!2 =50400
That is, there are 50400 ways of arranging the letter in the word STATISTICS.

例五.
A six-digit number is formed from the digits 1, 1, 2, 2, 2, 5 and repetitions are notallowed. How many these six-digit numbers are divisible by 5?

Solution
Then, the required number is 5!/2!3! = 10
That is, there are 10 of these six-digit numbers are divisible by 5.


Permutations
A permutation of a set of distinct objects is an ordered arrangement of these objects.An ordered arrangement of r elements of a set is called an r-permutation

例一.
Find the number of ways of placing 3 of the letters A, B, C, D, E in 3 empty spaces.

Solution
P(5, 3) = 5 × 4 × 3 = 60.

例二.
How many different ways are there to select one chairman and one vice chairman from a class of 20 students?

Solution
This is P(20, 2) = 20 × 19 = 380.

例三.

There are ten runners in a race and suppose that they have equal chance to win therace. The champion receives a gold medal, the runners-up receives a silver medal,and the second runners-up receives a bronze medal. How many different ways arethere to award these medals?

Hence, there are P(10, 3) = 10 × 9 × 8 = 720 possibleways to award the medals.