Cheat sheets
Tables you would otherwise count by hand: factorials, nCr, dice and card probabilities, sums of series, squares and cubes. Every number was computed by a program, not typed.
Factorials 0! to 15!
n! = n × (n-1) x ... x 2 × 1, and 0! = 1. Learn up to 8! by heart; the rest you can read off here. In exams you almost never multiply out a big factorial: cancel first. 10!/8! = 10 × 9 = 90.
| n | n! | Trick to remember |
|---|---|---|
| 0 | 1 | Defined as 1 (one way to arrange nothing) |
| 1 | 1 | |
| 2 | 2 | |
| 3 | 6 | |
| 4 | 24 | 24 hours in a day |
| 5 | 120 | 120 = 5 × 4! |
| 6 | 720 | 720 = degrees in 2 full turns |
| 7 | 5,040 | 5040 |
| 8 | 40,320 | 40,320 |
| 9 | 3,62,880 | 3,62,880 |
| 10 | 36,28,800 | 36,28,800 (about 36 lakh) |
| 11 | 3,99,16,800 | |
| 12 | 47,90,01,600 | |
| 13 | 6,22,70,20,800 | |
| 14 | 87,17,82,91,200 | |
| 15 | 13,07,67,43,68,000 |
- Cancel before you multiply: 12!/(10! x 2!) = 12 × 11 / 2 = 66.
- n! = n × (n-1)!, so 9! = 9 × 8! = 9 × 40,320 = 3,62,880.
- Trailing zeros of n! = floor(n/5) + floor(n/25) + floor(n/125)... e.g. 100! has 20 + 4 = 24 zeros.
Used in: Permutation and Combination · Probability
Combinations nCr: Pascal's triangle up to n = 12
nCr = n! / (r! (n-r)!) = the number of ways to CHOOSE r things from n when order does not matter. Each row of Pascal's triangle is nC0, nC1, ... nCn, and every number is the sum of the two above it.
Symmetry: nCr = nC(n-r). So 12C9 = 12C3 = 220; you never need r above n/2.
| n | r=0 | r=1 | r=2 | r=3 | r=4 | r=5 | r=6 |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||
| 2 | 1 | 2 | 1 | ||||
| 3 | 1 | 3 | 3 | 1 | |||
| 4 | 1 | 4 | 6 | 4 | 1 | ||
| 5 | 1 | 5 | 10 | 10 | 5 | 1 | |
| 6 | 1 | 6 | 15 | 20 | 15 | 6 | 1 |
| 7 | 1 | 7 | 21 | 35 | 35 | 21 | 7 |
| 8 | 1 | 8 | 28 | 56 | 70 | 56 | 28 |
| 9 | 1 | 9 | 36 | 84 | 126 | 126 | 84 |
| 10 | 1 | 10 | 45 | 120 | 210 | 252 | 210 |
| 11 | 1 | 11 | 55 | 165 | 330 | 462 | 462 |
| 12 | 1 | 12 | 66 | 220 | 495 | 792 | 924 |
- nC0 = nCn = 1, nC1 = n, nC2 = n(n-1)/2.
- Quick nCr: top r numbers of n! divided by r!. 10C3 = (10 × 9 × 8)/(3 × 2 × 1) = 120.
- Sum of a whole row: nC0 + nC1 + ... + nCn = 2n (number of subsets).
Used in: Permutation and Combination · Probability
Permutations nPr (ordered choices)
nPr = n!/(n-r)! = n × (n-1) x ... (r numbers multiplied). Use it when ORDER matters: ranks, seats, passwords, president/secretary. nPr = nCr × r!.
| n | r=1 | r=2 | r=3 | r=4 | r=5 |
|---|---|---|---|---|---|
| 1 | 1 | ||||
| 2 | 2 | 2 | |||
| 3 | 3 | 6 | 6 | ||
| 4 | 4 | 12 | 24 | 24 | |
| 5 | 5 | 20 | 60 | 120 | 120 |
| 6 | 6 | 30 | 120 | 360 | 720 |
| 7 | 7 | 42 | 210 | 840 | 2520 |
| 8 | 8 | 56 | 336 | 1680 | 6720 |
| 9 | 9 | 72 | 504 | 3024 | 15120 |
| 10 | 10 | 90 | 720 | 5040 | 30240 |
- Order matters? Use P. Order does not matter (team, committee, handshake, group)? Use C.
- 'Arrange all n' = nPn = n!.
Used in: Permutation and Combination
Counting shortcuts (no listing by hand)
Each line is a pattern that turns a 'count them' question into one calculation.
| Situation | Formula | Example |
|---|---|---|
| Arrange n different things in a row | n! | 5 books: 5! = 120 |
| Arrange with repeats (p alike, q alike...) | n! / (p! q! ...) | LETTER (T x2, E x2): 6!/(2! 2!) = 180 |
| Around a circle (n people) | (n - 1)! | 6 people: 5! = 120 |
| Necklace / garland (can flip it) | (n - 1)! / 2 | 7 beads: 6!/2 = 360 |
| Handshakes among n people | nC2 = n(n-1)/2 | 10 people: 45 |
| Diagonals of an n-sided polygon | n(n - 3)/2 | Hexagon: 9 |
| Triangles from n points (no 3 in a line) | nC3 | 8 points: 56 |
| Straight lines from n points (no 3 in a line) | nC2 | 8 points: 28 |
| Subsets of a set with n items | 2n | 5 items: 32 |
| Non-empty selections | 2n - 1 | 4 fruits: 15 |
| n identical things into r groups (any group may be empty) | (n + r - 1)C(r - 1) | 10 sweets, 3 kids: 12C2 = 66 |
| Same, every group gets at least one | (n - 1)C(r - 1) | 10 sweets, 3 kids: 9C2 = 36 |
| Letters in wrong envelopes (nobody right) | D(n): 0, 1, 2, 9, 44, 265 | 4 letters: 9 |
| Rectangles on an m × n grid of squares | (m+1)C2 × (n+1)C2 | Chessboard 8×8: 1296 |
| Squares on an n × n grid | 12 + 22 + ... + n2 | Chessboard: 204 |
| Numbers of k digits from n digits, repeats allowed | nk | 4-digit PIN: 104 = 10000 |
- Two tasks one after another (AND) → multiply. Either one task or the other (OR) → add.
- 'At least one' → total minus 'none'. Always faster than adding the cases.
Used in: Permutation and Combination · Probability · Counting Figures
Dice: every probability you need
Two dice give 6 × 6 = 36 equally likely outcomes. The number of ways to get a sum s is s - 1 for s up to 7, and 13 - s after that. So 7 is the most likely sum (6 ways).
| Sum of two dice | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Ways (out of 36) | 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 |
| Probability | 1/36 | 1/18 | 1/12 | 1/9 | 5/36 | 1/6 | 5/36 | 1/9 | 1/12 | 1/18 | 1/36 |
| Event with two dice | Ways | Probability |
|---|---|---|
| Doublet (both the same) | 6 | 1/6 |
| At least one six | 11 | 11/36 |
| No six at all | 25 | 25/36 |
| Sum is even | 18 | 1/2 |
| Sum is a prime (2, 3, 5, 7, 11) | 15 | 5/12 |
| Sum is a multiple of 3 | 12 | 1/3 |
| Sum greater than 9 | 6 | 1/6 |
| Product is even | 27 | 3/4 |
| Difference is exactly 1 | 10 | 5/18 |
Symmetric around 10.5: sum s has the same ways as 21 - s.
| Sum of three dice | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ways (out of 216) | 1 | 3 | 6 | 10 | 15 | 21 | 25 | 27 | 27 | 25 | 21 | 15 | 10 | 6 | 3 | 1 |
- One die: P(any given number) = 1/6, P(even) = 1/2, P(prime: 2, 3, 5) = 1/2.
- n dice: total outcomes 6n. P(at least one six in n throws) = 1 - (5/6)n.
Used in: Probability
Coins: exactly k heads
n coins give 2n outcomes. The number with exactly k heads is nCk. So P(exactly k heads) = nCk / 2n.
Divide the number in the cell by the outcomes. 4 coins, exactly 2 heads: 6/16 = 3/8.
| Coins | Outcomes | 0 heads | 1 heads | 2 heads | 3 heads | 4 heads | 5 heads | 6 heads |
|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 1 | 1 | |||||
| 2 | 4 | 1 | 2 | 1 | ||||
| 3 | 8 | 1 | 3 | 3 | 1 | |||
| 4 | 16 | 1 | 4 | 6 | 4 | 1 | ||
| 5 | 32 | 1 | 5 | 10 | 10 | 5 | 1 | |
| 6 | 64 | 1 | 6 | 15 | 20 | 15 | 6 | 1 |
- P(at least one head with n coins) = 1 - 1/2n.
- P(all same with n coins) = 2/2n.
Used in: Probability
A deck of 52 cards
4 suits × 13 ranks. Spades and clubs are black, hearts and diamonds are red. Face (court) cards are J, Q, K. Honour cards are A, J, Q, K (some papers include 10: say which you use).
| Group | How many | P(one card drawn) |
|---|---|---|
| Any one suit (e.g. hearts) | 13 | 1/4 |
| Red cards | 26 | 1/2 |
| Black cards | 26 | 1/2 |
| Aces | 4 | 1/13 |
| Kings | 4 | 1/13 |
| Face cards (J, Q, K) | 12 | 3/13 |
| Red face cards | 6 | 3/26 |
| Face cards or aces | 16 | 4/13 |
| Number cards 2 to 10 | 36 | 9/13 |
| King or heart (13 + 4 - 1) | 16 | 4/13 |
| Red king | 2 | 1/26 |
| Queen of spades | 1 | 1/52 |
Total ways to draw 2 cards: 52C2 = 1326.
| Two cards drawn together | Count | Probability |
|---|---|---|
| Both aces | 4C2 = 6 | 1/221 |
| Both red | 26C2 = 325 | 25/102 |
| Both the same suit | 4 × 13C2 = 312 | 4/17 |
| One red, one black | 26 × 26 = 676 | 26/51 |
| Both face cards | 12C2 = 66 | 11/221 |
Used in: Probability
Probability rules on one page
Four rules solve almost every placement probability question.
| Rule | Formula | Use it when |
|---|---|---|
| Basic | P(E) = favourable / total | All outcomes equally likely |
| Complement | P(not E) = 1 - P(E) | 'At least one', 'not all', 'none' |
| OR (addition) | P(A or B) = P(A) + P(B) - P(A and B) | Either can happen; subtract the overlap |
| OR, mutually exclusive | P(A or B) = P(A) + P(B) | They cannot happen together |
| AND, independent | P(A and B) = P(A) × P(B) | One does not affect the other (coins, dice, with replacement) |
| AND, dependent | P(A and B) = P(A) × P(B given A) | Without replacement: the second draw has one fewer |
| Conditional | P(A given B) = P(A and B) / P(B) | 'Given that...', 'if it is known...' |
| Odds in favour a : b | P = a / (a + b) | Odds against a : b means P = b / (a + b) |
| At least one of independent events | 1 - (1-p1)(1-p2)... | 'The problem is solved', 'at least one hits' |
| Exactly k successes in n tries | nCk pk (1-p)n-k | Repeated independent trials |
Used in: Probability
Summations and series
Never add a long list by hand. These formulas give the total at once.
| Sum | Formula | n = 10 | n = 20 | n = 50 | n = 100 |
|---|---|---|---|---|---|
| 1 + 2 + ... + n | n(n+1)/2 | 55 | 210 | 1,275 | 5,050 |
| 12 + 22 + ... + n2 | n(n+1)(2n+1)/6 | 385 | 2,870 | 42,925 | 3,38,350 |
| 13 + 23 + ... + n3 | [n(n+1)/2]2 | 3,025 | 44,100 | 16,25,625 | 2,55,02,500 |
| First n odd numbers 1 + 3 + 5... | n2 | 100 | 400 | 2,500 | 10,000 |
| First n even numbers 2 + 4 + 6... | n(n+1) | 110 | 420 | 2,550 | 10,100 |
| Series | nth term | Sum of n terms |
|---|---|---|
| Arithmetic (AP): a, a+d, a+2d... | a + (n-1)d | n/2 × (first + last) = n/2 [2a + (n-1)d] |
| Geometric (GP): a, ar, ar2... | a rn-1 | a (rn - 1)/(r - 1) |
| Infinite GP with |r| < 1 | - | a / (1 - r) |
| Average of an AP | - | (first + last)/2 = the middle term |
| Number of terms from a to l, step d | - | (l - a)/d + 1 |
- Sum of consecutive numbers = number of terms × average; the average is the middle term.
- Numbers from 1 to 100 divisible by 3: floor(100/3) = 33.
Used in: Number System · Series · Averages
Squares, cubes and powers
Squares to 30 and cubes to 15 save time on every paper. Powers of 2 show up in coins, subsets and binary questions.
| n | n2 | n | n2 | n | n2 |
|---|---|---|---|---|---|
| 1 | 1 | 11 | 121 | 21 | 441 |
| 2 | 4 | 12 | 144 | 22 | 484 |
| 3 | 9 | 13 | 169 | 23 | 529 |
| 4 | 16 | 14 | 196 | 24 | 576 |
| 5 | 25 | 15 | 225 | 25 | 625 |
| 6 | 36 | 16 | 256 | 26 | 676 |
| 7 | 49 | 17 | 289 | 27 | 729 |
| 8 | 64 | 18 | 324 | 28 | 784 |
| 9 | 81 | 19 | 361 | 29 | 841 |
| 10 | 100 | 20 | 400 | 30 | 900 |
| n | n3 | n | n3 | n | n3 |
|---|---|---|---|---|---|
| 1 | 1 | 8 | 512 | 15 | 3,375 |
| 2 | 8 | 9 | 729 | 16 | 4,096 |
| 3 | 27 | 10 | 1,000 | 17 | 4,913 |
| 4 | 64 | 11 | 1,331 | 18 | 5,832 |
| 5 | 125 | 12 | 1,728 | 19 | 6,859 |
| 6 | 216 | 13 | 2,197 | 20 | 8,000 |
| 7 | 343 | 14 | 2,744 | 21 |
| k | 2k | 3k | 5k |
|---|---|---|---|
| 1 | 2 | 3 | 5 |
| 2 | 4 | 9 | 25 |
| 3 | 8 | 27 | 125 |
| 4 | 16 | 81 | 625 |
| 5 | 32 | 243 | 3,125 |
| 6 | 64 | 729 | 15,625 |
| 7 | 128 | 2,187 | 78,125 |
| 8 | 256 | 6,561 | 3,90,625 |
| 9 | 512 | 19,683 | |
| 10 | 1,024 | 59,049 | |
| 11 | 2,048 | ||
| 12 | 4,096 | ||
| 13 | 8,192 | ||
| 14 | 16,384 | ||
| 15 | 32,768 | ||
| 16 | 65,536 |
- Square of a number ending in 5: n52 = n(n+1) then 25. 652 = 42|25 = 4225.
- A perfect square never ends in 2, 3, 7 or 8.
- Cube roots of perfect cubes: the last digit decides the last digit (2<→8, 3<→7 swap; the rest stay).
Used in: Multiply without long multiplication · Squares, cubes and roots in seconds · Number System
Fraction to percentage table
Know these and most percentage questions become one mental step.
| Fraction | Percent | Fraction | Percent |
|---|---|---|---|
| 1/1 | 100% | 1/11 | 9.09% |
| 1/2 | 50% | 1/12 | 8.33% |
| 1/3 | 33.33% | 1/13 | 7.69% |
| 1/4 | 25% | 1/14 | 7.14% |
| 1/5 | 20% | 1/15 | 6.67% |
| 1/6 | 16.67% | 1/16 | 6.25% |
| 1/7 | 14.29% | 1/17 | 5.88% |
| 1/8 | 12.5% | 1/18 | 5.56% |
| 1/9 | 11.11% | 1/19 | 5.26% |
| 1/10 | 10% | 1/20 | 5% |
- x% of y = y% of x. 16% of 25 = 25% of 16 = 4.
- A rise of 1/n then a fall back needs a fall of 1/(n+1).
Used in: Percentage · Profit and Loss · Discount
Unit digits of powers (cyclicity)
Only the last digit of the base matters. Find the exponent's remainder when divided by 4 (use 4 when the remainder is 0).
| Last digit | power 1 | power 2 | power 3 | power 4 | Repeats every |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 2 | 4 | 8 | 6 | 4 |
| 3 | 3 | 9 | 7 | 1 | 4 |
| 4 | 4 | 6 | 4 | 6 | 2 |
| 5 | 5 | 5 | 5 | 5 | 1 |
| 6 | 6 | 6 | 6 | 6 | 1 |
| 7 | 7 | 9 | 3 | 1 | 4 |
| 8 | 8 | 4 | 2 | 6 | 4 |
| 9 | 9 | 1 | 9 | 1 | 2 |
- 795: 95 leaves 3 when divided by 4, so the unit digit is the same as 73 → 3.
- 0, 1, 5, 6 always end in themselves.
Used in: Number System
Units, conversions and exam-hall tactics
Small conversions cause most silly mistakes. Keep these next to you.
| Convert | How |
|---|---|
| km/h to m/s | x 5/18 (72 km/h = 20 m/s) |
| m/s to km/h | x 18/5 (15 m/s = 54 km/h) |
| 1 hectare | 10,000 m2 |
| 1 litre | 1000 cm3 |
| 1 m3 | 1000 litres |
| 1 km | 1000 m |
| 1 hour | 3600 seconds |
| π | 22/7 (use when a radius is a multiple of 7), else 3.14 |
| Under time pressure | How it saves time |
|---|---|
| Look at the options first | Often only one option has the right unit digit or size |
| Unit-digit check | Multiply only the last digits to rule options out |
| Digit-sum (casting out 9s) check | Digit sums of the question and answer must match |
| Approximate, then pick | Round 49.8% to 50%; pick the option nearest your estimate |
| Plug the options in | For ages and 'find the number', test the middle option first |
| Take 100 or the LCM | Turns percentages and work problems into whole numbers |
| Skip and return | Spend at most 90 seconds on a question in the first pass |
Used in: Time and Distance · Problems on Trains · Boats and Streams · Mensuration