# SOLVED PROBLEMS ON COMBINATIONS

Solved Problems on Combinations :

In this section, we will learn, how to solve problems on combinations.

## Solved Problems on Combinations

Problem 1 :

There are 5 teachers and 20 students. Out of them a committee of 2 teachers and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees

(i) a particular teacher is included?

(ii) a particular student is excluded?

Solution :

Number of ways in which the commitee can be formed  =

5C2   20C3

(i) a particular teacher is included?

When a particular teacher always included, it is enough to select the remaining 1 teacher out of 4 teachers.

Total number of ways  =  =  4C1   20C3.

(ii) a particular student is excluded?

When a particular student always excluded, it is enough to select the remaining 2 students out of 19 students.

Total number of ways  =  =  5C2   19C3.

Problem 2 :

In an examination a student has to answer 5 questions, out of 9 questions in which 2 are compulsory. In how many ways a student can answer the questions?

Solution :

Total number of questions  =  9

Number of questions must be answered  =  2

Number of questions to be answered  =  5

Number of ways to answer the questions  =  9 - 2  =   7C3

=  35

Problem 3 :

Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly three aces in each combination.

Solution :

Total number of cards in a deck  =  52

Number of ace cards  =  4

Number of cards to be selected  =  5

Here we must select 3 ace cards out of 4 and 2 other cards out of 48.

Number of ways  =  4C  48C2

=  4 ⋅ [(48 ⋅ 47)/2]

=  4 (1128)  =  4512

Hence the total number of ways is 4512.

Problem 4 :

Find the number of ways of forming a committee of 5 members out of 7 Indians and 5 Americans, so that always Indians will be the majority in the committee.

Solution :

 7 Indians 5 Americans Committee to be formed 543 012 555

Number of ways  =  (7C  5C0) + (7C  5C1) + (7C  5C2)

=  (21 ⋅ 1) +  (35 ⋅ 5)  + (35 ⋅ 10)

=   21 + 175 + 350

=  546

Hence the required number of ways is 546.

After having gone through the stuff given above, we hope that the students would have understood, how to solve problems on combinations

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

You can also visit our following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6