If one operation can be performed in m ways and another in n ways, both together can be performed in m × n ways.
👉 Use when more than one task needs to be done in sequence.
If only one operation out of several is to be performed, and one can be done in m ways and another in n ways, then total ways = m + n
👉 Use when only one out of the possible tasks is done.
A permutation is an arrangement of n objects taken r at a time in a specific order.
nPr = n! / (n - r)!
For n distinct objects: n!
With repetition allowed: n^r
With p, q, r objects of same kind: n! / (p! q! r!...)
Distinct objects around a circle: (n – 1)!
Necklace/Garland (reflection counts as same): (n – 1)! / 2
k consecutive items around a circle:
If k < n → n
If k = n → 1
n items along r-sided regular polygon with n/r items per side = n! / r
If polygon is not regular: use permutation → nPr = n! / (n - r)!
Repetition allowed in r places: n^r
People around a rectangular table with equal numbers on each side: n! / 2
Derangement (No item in original position):
Values:
D(2) = 1
D(3) = 2
D(4) = 9
D(5) = 44
A combination is the selection of r objects from a set of n without considering the order.
nCr = n! / [r! (n - r)!]
n+r-1Cr
Distribute n identical items among r groups:
With zero allowed: n+r-1Cr-1
At least one per group: n-1Cr-1
a) nC0 = 1 = nCn
b) nC1 = n = nCn-1
c) nCr = nCn-r
d) nCa = nCb, if a + b = n
e) nCr + nCr-1 = n+1Cr
f) ∑nCr = 2^n
g) Sum of even nCr = sum of odd nCr = 2^(n-1)
Without repetition:
Sum = (Sum of digits) × (111… up to r digits) × (nPr / n)
With repetition:
Sum = (Sum of digits) × (111… r digits) × n^(r - 1)
4 letters in 7 boxes (different) = 7^4 = 2401
4 identical rings on 5 fingers = 5^4 = 625
4 different rings on 5 fingers (repetition allowed) = 5 × 6 × 7 × 8 = 1680
Lines from n points (no 3 collinear): nC2
Triangles from n points: nC3
Diagonals in polygon = nC2 – n
If groups are distinct: = n! / (p! q! r!...)
If groups are indistinct but equal size: = n! / [(n/r)!]^r = If groups not distinct → divide further by r!
| Concept | Formula |
|---|---|
| Linear Permutation (no repeat) | nPr = n! / (n - r)! |
| Linear Permutation (repeat) | n^r |
| Identical items in permutation | n! / (p! q! …) |
| Circular Arrangement | (n - 1)! |
| Necklace/Garland | (n - 1)! / 2 |
| Combination (no repeat) | nCr = n! / [r!(n - r)!] |
| Combination (with repeat) | n + r - 1Cr |
| Distribution of identical items | n + r - 1Cr - 1 |
| Derangements | Dn = n! × [1 – 1/1! + 1/2! – … + (-1)^n/n!] |
| Lines from n points | nC2 |
| Triangles from n points | nC3 |
| Diagonals in polygon | nC2 – n |