Combinations Concept and Problems to crack CAT Quantitative Aptitude

Combinations

The number of different groups of r objects that can be formed from a total of n objects when the order of selection is irrelevant is called the number of combinations of n objects taken r at a time.  

The number of these r combinations is denoted by ⁿC_rorC(n, r)and

Recall that ⁿP_r is the number of ways in which r objects can be selected from n objects, when the order of selections is relevant.

We can also choose an ordered selection of r object out of n objects in two steps.

1. Choose a subset consisting of r objects without paying any attention to the order in which they are chosen. This can be done in ⁿC_r ways.

2. For a given choice of r objects, count the number of ways in which these object could be ordered. Any r objects can be permuted amongst them in r! ways.

Hence from the fundamental rule of counting we must have :

The second and the third equalities follows from the fact that:

Example:

(a) In a game of poker a player receives 5 cards from an ordinary deck of playing cards. These 5 cards are called a poker hand. How many poker hands are there?    

(b) A poker hand is said to be a flush if all the cards are of the same suit. In how many ways can one be dealt a flush?

Solution:

(a) The number of poker hands that a player can receive is

(b) There are four suits in a deck of cards. Out of these 4 suits, one suit has to be chosen and from this suit 5 cards have to be chosen (out of 13 cards in the four suits). The number of ways one can be dealt a flush is:

Example:

(a) A student has to answer 6 out of 10 questions in an examination. How many choices does she have?

(b) How many choices are there if she must answer at least 2 of the first 3 questions?

Solution:

(a) The student has to select 6 out of 10 of questions. There are

choices for her.

(b) If the student must answer at least 2 of the first 3 questions, it is convenient to divide her choices into tow mutually exclusive groups.

(i) She answers all three of the first three questions and answers any 3 of the remaining 7 questions. This can be done in

(ii) She chooses 2 out of the first 3 questions and answers any 3 of the remaining 7 questions. This can be done in

Hence the number of choices that are available to the student is

35 + 105 = 140

Permutations and Combinations is a topic which is applied together. Read an article on Permutation to learn Permutations and Combinations fully.