Combinations

Just like permutations, combination is also a way of selecting objects from a larger group, but without considering the order of objects. A combination is thus a selection of r items from a total of n items where order hasn’t got any importance. The formula used for this computation is

nCr = n!/r!(n-r)!


CombinationCombinations can be of two types:
  • Combinations in which there is no repetition
  • Combinations in which repetition is allowed

Combinations with repetition are a bit confusing and so we first consider the case of combinations without repetition which are comparatively simple.

Selection of objects without repetition:

The number of selections (combinations or groups) that can be formed from n different objects taken r (0 < r < n) at a time is nCr = n!/(r!(n-r))!.

Explanation: Let the total number of selection (or groups) = x. Each group contains r objects, which can be arranged in r! ways. Hence number of arrangements of r objects = x × (r!).

But the number of arrangements = npr

=> x × (r!) = nPr 

=> x = nPr/ r!

=> x = n!/(r!(n-r)!) = nCr.

Selection of objects with repetition:

If we need to compute the combination of n different items taken r at a time wherein the possibility of occurrence of any item may be once, twice or thrice….. up to r times, then it is calculated with the help of the formula n+r-1Cr.

Explanation: Let us consider n objects a1, a2, a3, ...... , an. Now suppose we have a collection of r objects in which

a1 occur x1 times,

a2 occur x2 times,

a3 occur x3 times,

.................................

.................................

an occur xn times,

such that x1 + x2 + x3 + ......... + xn = r                   ......... (1)

0 < xi < r ∀ i ∈ {1, 2, 3, ......, n}.

Now the total number of selections of r objects, out of n

= number of non-negative integral solution of equation (1)

n+r-1Cn-1 

n+r-1Cr.

Let us refer some of the examples to see what kind of questions can be asked from these sections:

Illustration: Let 15 toys be distributed among 3 children subject to the condition that any child can take any number of toys. Find the required number of ways to do this if 

(i) toys are distinct                               (ii) if toys are identical

Solution:

(i) Toys are distinct

Here we have 3 children and we want the 15 toys to go to the 3 children with repletion. In other words it is same as selecting and arranging children 15 times out of 3 children with the condition that any children can be selected any no. of time which can be done in 315 ways (n = 3, r = 15).

(ii) Toys are identical

Here we only have to select children 15 times out of 3 children with the condition that any children can be selected any number of times which can be done in

3+5-1C15 = 17C2 way (n = 3, r = 5).

Illustration: in how many ways can a pack of 52 cards be divided equally amongst four players in order? Also find the ways of dividing it into four groups of 13 cards each.

Solution: Let A, B, C and D be the four players and since the 52 cards are to be distributed equally amongst these players, then each will get 13 cards.

Now, 13 cards can be given to A in 52C13 ways. Out of the remaining 39 cards, 13 cards can be given to B in 39C13 ways. Hence, similarly C will get 13 cards in 26C13 ways.

The remaining 13 cards will be given to D. Hence, the total number of ways in which a pack of 52 cards can be distributed equally amongst 4 players in order is given by

= 52C13 x 39C13 x 26C13 x 13C13

= 52! /39! 13! x 39!/26!13! x 26!/13!13! x 1

= 52!/(13!)4

Similarly, the number of ways in which a pack of 52 cards can be divided equally into 4 groups of 13 cards each

= 1/4! x 52C13 x 39C13 x 26C13 x 13C13

= 52!/4! (13!)3

View the video for more on the topic 

Illustration: Find the total number of ways in which six ‘+’ and four ‘-‘ signs can be arranged in a line such that no two ‘-‘ signs occur together.

Solution: Six ‘+’ can be arranged in a line in just one way and in that case they create 7 places shown in the form of dots below:

.+.+.+.+.+.+.

These dots are negative signs and 4 out of these 7 dots may be chosen in 7C4 ways

= 7.6.5.4/ 1.2.3.4.

= 35 ways

 

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