Restricted Combination, Illustration And Examples, Important Topics For JEE Maths 2024

Restricted Combination : Combination was first used in problem 79 in 16 ^th century BCE. It involves geometric series similar to Fibonacci’s problem for counting composition of 1 and 2 giving a particular sum. Combination denotes number of selections means if a child has to select 3 toys from a set of 4 different toys A, B, C, D than total number of selections would be and could be denoted as ABC, ABD, ACD, BCD during combination arrangements of objects does not count restricted combination means if any restriction is imposed like some particular objects cannot be selected or some particular objects must be selected

, some particular objects must be together or some particular objects cannot be together, any particular object must be in any particular position etc. any kind of restriction always reduces number of outcome as compare to the total number of outcomes without restriction.

for example, selection of 4 objects out of 6 different objects would be with a value equal to 15 but if a restriction is being imposed such as 1 particular object must not be selected than number of ways would be which is equal to 5 and if a particular object must always be in the selection than number of ways would be this is because selection of particular object must be from it only as and remaining 3 objects could be selected from 5 objects as  in ways and since these events are independent and individual subparts hence outcomes must be multiplied.

We can categories restricted combination in various parts such as

• Object can be included or excluded in selection
• Some objects must be together or not
• Some objects must be in a fixed position always

Let’s explore above restriction with examples

Examples based on restricted combination

Example 1: Find total number of ways of making a cricket team of 11 players consisting of 6 batsman, 4 bowlers and 1 wicket keeper from 10 batsman, 6 bowlers and 2 wicket keepers having 2 particular batsmen always in team?

Sol. Restriction is two particular batsmen must always be there so selection of two particular batsmen would be as now remaining 9 players must consist of 4 batsman, 4 bowlers and 1 wicket keeper, which can be done in ways now total number of ways would be = 2100 ways

Example 2: Find total number of ways word “TRIANGLE” could be arranged keeping vowels together?

Sol. TRIANGLE consist of 3 vowels as IAE and consonants as TRNGL now to keep vowels together we can use Tie method which means to tie the objects which has to be together and count them as single one so here we have ‘IAE’ and ‘T’, ‘R’, ‘N’, ‘G’, ‘L’ total 6 objects which could be arranged in 6! Ways now ‘IAE’ could also be arranged in 3! Ways so total number of arrangements would be 6!

Example 3 : Find total Number of arrangements 6 students in a row having 4 boys and 2 Girls, while keeping Girls in odd places?

Sol. Number of odd places we have as 1,3,5 and 2 of them can be selected as Now these girls could arrange mutually in 2! Ways and others 4 students could arrange in remaining 4 places in 4! Ways Now total ways would be ways

Rapid Questions (Based On Restricted Combination)

1. Find number of ways of forming a committee of 8 persons having 1 chairperson, 1 vice chairperson, 1 secretary and 5 others form 10 peoples keeping chairperson eldest while no person has same age in group?

2. Find number of ways of arranging word ‘EQUATION’ keeping all vowels together?

Complex illustrations on restricted combinations

Q.1 : How many words can be formed from letters of the word ‘MONDAY’ starting with ‘O’?

Sol. Restriction here is that first word must start with O hence first position has 1 way to be filled and in remaining position letters could be arrange in 5! Ways total ways equal to 120.

Q.2: How many ways can a lawn tennis mixed double of opposite gender couples be arranged from 9 married couples if no husband and wife play in same set?

Sol. Mixed double means each side must contains two persons we have 9 marries couples means 9 husband and 9 wives Now let select 2 husbands out of 9 husbands in ways now since husband and wife can’t be in the same set and match has to be between married couples only hence remaining two players must be selected from 7 women’s as ways now match between them could be arranged in in 4 ways this could understand as let two husbands are , and wives are now match could be arranged as so now total possible matches would be 4=3024

Rapid Questions (Based On Restricted Combination)

1. How many ways can a lawn tennis mixed double of opposite gender couples be arranged from 5 married couples if no husband and wife play in same set?

2. How many words can be formed from letters of the word ‘FRIDAY’ starting with ‘I’