Permutation and Combination

PROBLEM 1

How many possible ways are there of seating seven people, P, Q, R, S, T, U and V at a circular table  (U.M.E 2005, Question 37)

 

SOLUTION

The possible arrangement around a circular table is given by (n – 1)!

Number of people = 7

Possible arrangement

 

PROBLEM 2

Five people are to be arranged in a row for a group photograph. How many arrangement are     there if a married in the group insist on sitting next to each other (U.M.E )

 

SOLUTION

Take the couple as one

The number of way 4 people can be arranged is ⁴P₄ = 4ways

The couple can be seated in 2ways. Therefore, the possible arrangement is

 

PROBLEM 3

PROBLEM 4

How many two – digits number can be formed from digits 0, 1, 2, 3, if a digits can be    repeated and no digit number may begin with 0 (U.M.E 1999, Question 48)

 

SOLUTION

The first digit can be formed in ³P₁ = 3 ways   (since zero must not start)

The second digit can be formed in ⁴P₁ = 4 ways

The possible formation is    

           

PROBLEM 5

In how many ways can a delegation of 3 be chosen from among 5 men and 3 women, if at         least one man and at least one woman must be included? (U.M.E 2000, Question 47)

 

PROBLEM 6

In how many ways can the word MATHEMATICS be arranged (U.M.E 2000, Question 48)

 

SOLUTION

The total number of letters that formed MATHEMATICS is 11 letters

Letter that are repeated are A, M , T   (2A, 2M, 2T)

PROBLEM 7

Find the number of ways of selecting 8 subjects from 12 subjects for an examination (U.M.E 2001,Question 49)

 

SOLUTION

            PROBLEM 8

How many three –digit number can be formed from 32564 without being repeated (U.M.E        2002, Question 47)

 

SOLUTION

PROBLEM 9

SOLUTION

 

 

PROBLEM 10

Find the number of committees of three that can be formed consisting of two men and one

woman from four men and three women (U.M.E 2003, Question 46)

 

 

PROBLEM 11

In how many ways can 2 students be selected from a group of 5 students in a debating competition? (U.M.E 2004, Question 22)

 

SOLUTION

PROBLEM 12

A committee of six is to be formed by a state governor from state commissioners and three        members of the state house of assembly. In how many ways can the members of the      committee be chosen so as to include one member of the house of assembly? (U.M.E 2004,           Question 25)

 

SOLUTION

Since one of the members of the house of assembly is to be included in the committee, five         more people will need to be selected from the nine state commissioners and two members of       the house of assembly.

 

The possible selection will be

 

PROBLEM 13

In how many ways can the letters of word ACCEPTANCE be arranged?

            

(UME 2008, Question 47 set U03, type A)

 

SOLUTION

The word “Acceptance” consist of 10 letters We have 3C_s, 2A_s, 3E_s

Possible Arrangement =

Anwser: Option A

 

PROBLEM 14

Find the number of ways of selecting 6 out of 10 subjects for an examination

(a)  218   (b)  216   (c)  215   (d)  210 (UME 2008, Question 48 set U03, type A)

 

SOLUTION

Possible ways of selection

Answer: Option D

           

            PROBLEM 15

In how many ways can a delegation of 3 be chosen from 5 men and 3 women, if at least 1         man and 1 woman must be included? (UME 2009, Question 47 set U03, type A)

           

            Solution

           

            PROBLEM 16

            In how many ways can 9 people be seated if 3 chairs are available?

            (A) 720   (B) 504   (C)  336   (D)  210 (UME 2009, Question 48 set U03, type A)

 

Solution

Answer Option B