Permutation and Combination of Alike Objects for JEE

Permutation and combination are key ideas in mathematics that we employ to choose data or objects from a given set. Permutations and combinations are collections of things that can be used together to form a group. The mathematical ideas of permutation and combination were initially introduced by Blaise Pascal and combination- one of the essential topics in mathematics, is used in science, engineering, and research. The distinction is that  permutation is the order in which the data is placed is known,  whereas combination is when a set of information is selected from a specific group.

Permutation and Combination

Permutation: 

The term 'arrangement' can be seen in the definition of permutation. Permutation refers to the simultaneous arrangement of some or all of the same or different things. In this context, the meaning of arranging is choosing and managing.

Combination: 

Combination refers to a set of elements that may be similar or dissimilar and are related to choosing one or more items. A combination can also refer to the collection, selection, or committee of words.

Permutation and Combination Formulas

• For Permutation

'r'(n !){(n-p)!

Example- All conceivable combinations using the letters a, b, and c.

By taking two at a time are ab, ac, ba, bc, ca, cb.
By taking all three at a time are abc, acb, bac, bca, cab, cba.

• For Combination

There are various options for selecting 'r' items from 'n' items. 

${ }^{n}C_{r}=\dfrac{n !}{r !(n-r) !}$ 

Example- All the potential combinations using the letters a, b, and c. 

When three letters out of three are to be chosen, abc is the sole option.
When selecting two out of three letters, the possible combinations are ab, bc, ab.

Some Other Formulas

• If n is even ⁿC_r is greatest for $r=\dfrac{n}{2}$

• ${ r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}$
• ${ }^{n} C_{r}= { }^{n} C_{n-r}$

Difference Between Permutation and Combination

When things are of a distinct type, permutation refers to the various conceivable arrangements. To understand how to use permutation and combination correctly, you must first understand the difference. We only care about the set of items that make up a particular group in combinatorics; conjunction refers to the number of smaller groups or locations that can be constructed from the constituents of a more extensive set. Refer to the table below to better understand the difference between permutation and combination.

A permutation's response is always greater than a combination's answer. The permutation value is always more significant than the mix value for given values of n and r. Different alternative arrangements are counted in permutations, whereas only different subgroups are counted in combinations.

Permutation and Combination Examples

Example 1: Determine the number of distinct ways the word 'ASSASSINATION' can be spelt.

Solution:  

We've offered 13 alphabets in this example.

As a result, the number of possible permutations is enormous $=\dfrac{13 !}{3 ! 4 ! 2 ! 2 !}$

Example 2: Determine the number of words made up of 5 letters from the letters in the word "INDEPENDENCE."

Solution: 

We've offered 12 alphabets in this example

Case 1 - When EEEDC (2 different and 3 alike)

Number of selections $={ }^{2} C_{1} \times{ }^{5} C_{2!}{3!}$

Thus, number of permutation $=20\times\dfrac{5!}{3!}=400$

Case 2 - When EEEEP (4 alike and 1 different)

We only have one option for four people who are the same, namely E, and five alternatives for choosing one person who is different. 

Number of selection $=1\times { }^{5} C_{1}=5$ 

Arrangement $=\dfrac{5!}{4!}$

Thus, 

Case 3 - When EENDI (2 alike and 3 different)

Number of selection $= {}^{3} C_{1}\times{ }^{5} C_{3}=30$

Arrangement $= {}^{2} C_{1}\times{ }^{5} C_{1}=4$ 

Arrangement $= \dfrac{5!}{3!2!}$

Thus, the number of permutations $= 4\times\dfrac{5!}{3!2!}=40$

Case 4 - When EDIPC (All five different)

Number of selection $= { }^{6} C_{5}=6$

Arrangement END (2 alike, 2 other alike and 1 different)

Number of selection $= { }^{3} C_{2}\times4=12$

Arrangement $= \dfrac{5!}{2!2!}$

Thus, the number of permutations $= 12\times\dfrac{5!}{2!2!}=360$

As a consequence, sum all of the outcomes from the various situations together is:

$= (400+ 25 + 1800 + 40 + 720 + 360)$

$= 3345$

Conclusion

The significant difference between permutation and combination in  mathematical notions is point order, placement and position, essential in permutation but not in combinations. Combination arranging objects, persons, numbers, letters, colors, and so on.

A combination is an unordered set or pair of values within the defined criteria. A single combination can provide a large number of permutations. A single permutation, on the other hand, yields only one combination. Following the above discussion, it is evident that permutation and combination are two different words used in mathematics, statistics, research, and everyday life.