Venn diagrams, arrangements, selections and probability are all one skill: drawing the situation so you can count it. Press Play and watch each picture build step by step, or use the arrow keys. Turn on Voice to have it read aloud. Each chapter ends with the CAT trap and the fastest shortcut.
Everything from the player, compressed. Read it once before each practice burst.
| Situation | Picture → rule | Example |
|---|---|---|
| Sets and Venn diagrams | ||
| Any Venn question | Fill from the centre outwards: centre → pair-only rooms → single rooms → outside | pair 15, centre 5 → pair-only 10 |
| Two sets | A + B − both | 40 + 30 − 12 = 58 |
| Three sets | A+B+C − (pair totals) + centre | 120 − 37 + 5 = 88 |
| Exactly one / two / three | A+B+C = E1 + 2·E2 + 3·E3; people = E1+E2+E3 | 120 = 61 + 44 + 15 |
| Two-set overlap range | max = smaller set; min = A + B − total (or 0) | 60, 50 in 80 → 30 to 50 |
| Max / min of the centre | Subtract: E2 + 2·E3 = (A+B+C) − people. Max → E2 as small as allowed. Min → E1 as small as allowed. Then fill and check every room | 215 − 100 = 115 → 57 and 15 |
| Counting | ||
| Choices made together / alternatives | AND → multiply · OR → add | 3 × 2 × 2 = 12 outfits |
| Order matters (arrange r of n) | Slots: n × (n−1) × … = nPr | 5 × 4 × 3 = 60 |
| Order doesn't matter (select r of n) | nCr = nPr ÷ r! · nCr = nC(n−r) | 60 ÷ 6 = 10 |
| Repeated items | n! ÷ (p! q! …) | BANANA: 720 ÷ 12 = 60 |
| Always together | Glue into one block × inside orders | 4! × 2 = 48 |
| Never together | total − together, or seat the others and use gaps | 120 − 48 = 72 |
| Round table / necklace | (n−1)! · necklace (n−1)! ÷ 2 | 5 people: 24 · beads: 12 |
| Identical items to r people | Dots and bars: (n+r−1)C(r−1); each ≥ 1: hand out first | 7 to 3: 36 · each ≥1: 15 |
| Grid paths (right / up) | A word of R's and U's: (a+b)Cb | 4 × 3 grid: 35 |
| Handshakes, lines, diagonals | nC2 · diagonals nC2 − n | 10 people: 45 · octagon: 20 |
| Digits with conditions | Fill the fussiest slot first; split cases on 0 | 3-digit even, distinct: 328 |
| Probability | ||
| Equally likely outcomes | favourable squares ÷ all squares | sum 7 on 2 dice: 6/36 |
| Many "yes" cases | 1 − P(not) | ≥ one six: 1 − 25/36 |
| Independent events | Multiply along a tree path, add across paths | 1/2 × 1/3 = 1/6 |
| At least one in n tries | 1 − (1 − p)ⁿ | 4 rolls: 1 − (5/6)⁴ ≈ 0.52 |
| Without replacement | Shrink the bag after each draw, or nCr ratios | 5/8 × 4/7 = 5/14 |
| Expected value | Balance point: Σ value × probability | die: 3.5 |
Three small benches. Each one is a picture from the player that you control.
100 people, each in at least one of P (70), C (65), M (80). Change the rooms until every circle total matches, then try to make the centre as big, then as small, as possible. Tap a room in the picture to pick it.
Share 5 identical sweets among Asha, Bala and Chetan. Tap any spot to make it a bar; exactly 2 bars are allowed. Find every different sharing.
Pick an event. The board colours the favourable squares. Switch on "not" to see the other side, which is often quicker to count.
Written for this page, easy to CAT level. Each tag names the picture to draw. Give yourself about 2 minutes each; draw first, then calculate.
In a class of 50, 32 play chess, 25 play carrom and 9 play neither. How many play both?
Two dice are rolled. What is the probability that the product of the two numbers is even?
A colony has 200 households. 90 read paper A, 80 read B and 70 read C. 30 read A and B, 25 read B and C, 20 read A and C, and these figures include the 10 that read all three. How many read exactly one paper, and how many read none?
In how many arrangements of the letters of SUCCESS are the two C's next to each other?
A committee of 4 is chosen at random from 6 men and 4 women. What is the probability that it has at least one woman?
How many 4-digit numbers with all digits different are divisible by 5?
Six people sit around a round table. In how many seatings are two particular people, X and Y, not next to each other?
How many integer solutions does a + b + c = 15 have, with a ≥ 1, b ≥ 2 and c ≥ 3?
You walk from (0, 0) to (5, 4), one unit right or one unit up at each step. How many paths do not pass through (2, 2)?
120 people each speak at least one of Hindi, Tamil and Bengali. 80 speak Hindi, 70 Tamil and 60 Bengali. Each of the seven possible combinations of languages is spoken by at least one person. What is the largest possible number who speak all three?