Correct Answer - Option 3 :
\(\rm \frac{10!}{2!\;\times \;2!}\)
Concept:
- The ways of arranging n different things = n!
- The ways of arranging n things, having r same things and rest all are different = \(\rm n!\over r!\)
- The no. of ways of arranging the n arranged thing and m arranged things together = n! × m!
- The number of ways for selecting r from a group of n (n > r) = nCr
- To arrange n things in an order of a number of objects taken r things = nPr
Calculation:
The total number of words in GIRLFRIEND is 10
The word "I" in GIRLFRIEND repeated twice
Also, the word "R" GIRLFRIEND repeated twice
So, Number of different permutations = \(\rm \frac{10!}{2!\times 2!}\)
Permutation: Permutation is a way of changing or arranging the elements or objects in a linear order.
The number of permutations of 'n' objects taken 'r' at a time is determined by the following formula:
nPr = \(\rm \frac{n!}{(n - r)!}\)
nPr = permutation
n = total number of objects
r = number of objects selected
The factorial function (Symbol: !) just means to multiply a series of descending natural numbers.
For examples:
4! = 4 × 3 × 2 × 1
1! = 1
There are three types of permutation:
- Permutations with Repetition
- Permutations without Repetition
- Permutation when the objects are not distinct (Permutation of multi-sets)
Representation of Permutation:
We can represent in many ways such as:
- P (n, k)
- \(\rm P_{k}^{n}\)
-
n Pk
-
n Pk
- P n, k
Application of Permutations:
- Permutations are important in a variety of counting problems (particularly those in which order is important).
- Permutations are used to define the determinant.
Order is very important in permutation.
"A Permutation is an ordered combination."
Permutation |
Combination |
Permutation means the selection of objects, where the order of selection matters |
The combination means the selection of objects, in which the order of selection does not matter. |
In other words, it is the arrangement of r objects taken out of n objects. |
In other words, it is the selection of r objects taken out of n objects irrespective of the object arrangement. |
The formula for permutation is nPr = \(\rm \frac{n!}{(n - r)!}\)
|
The formula for combination is
nCr = \(\rm \frac{n!}{r!(n - r)!}\)
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