Concept and definition.
Factorial of a number (whole number only) is equal to the product of all the natural numbers up to that number. Factorial of n is written as n or n! and is read as factorial n.
Hence 7! = 7 ×6 × 5×4 × 3 × 2 ×1 = 5040
5! = 5 ×4 ×3 ×2×1 = 120
n! = n× (n – 1)× (n – 2) × …… 3 × 2×1
1]0! = 1 (by definition)
3] 2! = 2
4] 3! = 6
5] 4! = 24
6] 5! = 120
7] 6! = 72
Permutation or combination is the number of ways in which an event occurred.
AND denotes Multiplication ,
OR denotes Addition
1. We labelled each block with the number of ways, options, or alternatives available.
2. We multiply the number of blocks in each row.
3. In the case of code repetition always allow
4. The first digit in a code could be zero.
5. In Number, it is not possible to begin with zero.
The question should specify whether there will be a repetition or not in the answer.
1. Permutations of n different things taken ‘r’ at a time is denoted by nPr and is given by nPr = (n! )/((n – r)! )
where r ≤ n. When to use it?
When n distinct items present and r have to be selected and then arranged.
Arrangements[permutation] – keywords – seating, sitting, sequence, order, alphabets, schedule, ranking, itinerary, codes, numbers, rows, lines, position.
Order important – gives unique arrangements
For e.g. A and B sitting on a chair can be AB or BA so these are two distinct arrangements
It is basically selection followed by arrangement.
So nPr =n!/(n-r)!
The number of permutations when things are not all different.
If there be n things, p of them of one kind, q of another kind, r of still another kind and so on, then the total number of permutations is given by (n! )/(p! q! r!..)
Ex: In how many ways can you arrange the letters of the word Banana?
5 people A, B, C, D, E to be arranged in which A and B are together.
4 ! × 2!
5 people A, B, C, D, E to be arranged in which A and B are not together.
5! – 4! × 2!
1) No. of circular permutations of n things taken all at a time = (n – 1)!
(2)Circular Permutations: ((n – 1) !)/(2 )
(if clockwise and anti-clockwise doesn’t matter )
No. of circular permutations of n different things taken r at a time = nPr/r
- Number of combinations of n dissimilar things taken ‘r’ at a time is denoted by nCr & is given by nCr = (n! )/((n – r)! r!) where r ≤ n
- Number of combinations of n different things taken r at a time in which q particular things will always occur is
n – qCr – q
- No. of combinations of n dissimilar things taken ‘r’ at a time in which ‘q’ particular things will never occur is
n – qCr
- nCr = nCn – r
Combibation[Selection]- keywords – team, committee, balls, handshakes, matches, picking, random, select choose.
Order not important – For example choosing A and B from a group of 3 or four alphabets.
The order does not matter.
India playing a match against Australia is the same as Australia playing against India.