CAT — Modern Math: Permutation & Combination, Probability, Set Theory

Why Modern Math matters for CAT

Modern Math tests discrete thinking, counting strategies, probability modeling, and set manipulations. These topics reward pattern recognition, complement principle, symmetry, and algebraic counting (generating functions rarely, but combinatorial identities often). We'll emphasise diagram-first approaches (Venn for sets, tree for probability, slots for permutations).

Permutations & Combinations

Key formulas: Permutations: P(n,r)=nPr = n!/(n−r)!. Combinations: C(n,r)=nCr = n!/(r!(n−r)!). nCr = nC(n−r).

Tricks: Use complement (total − unwanted), treat identical objects using division by factorials, and arrange with slots and separators for distributions. For circular permutations use (n−1)! when rotations identical.

Solved examples

Example: Number of ways to choose 3 people from 8 = C(8,3)=56. Ways to arrange them in order = P(8,3)=336.

Example (identical objects): Permutations of letters in 'LEVEL' = 5!/(2!2!)=30.

Probability

Basics: For finite sample space, P(E) = favourable/total. For independent events A,B: P(A∩B)=P(A)P(B). Conditional: P(A|B)=P(A∩B)/P(B). Bayes' rule for reversals.

Tricks: Draw tree diagrams for sequential experiments; use complement for "at least one" events; for non-uniform sample spaces weight by counts; linearity of expectation for expected value problems (no independence required).

Solved examples

Example: Probability of at least one head in 3 fair coin tosses = 1 − P(no heads) = 1 − (1/2)^3 = 7/8.

Example (conditional): Two cards drawn without replacement: P(second is ace | first is ace) = 3/51 = 1/17.

Set Theory

Basics: Union & intersection: |A∪B| = |A| + |B| − |A∩B|. For three sets use inclusion-exclusion: |A∪B∪C| = sum singles − sum pairs + triple intersection.

Tricks: Venn diagrams are essential; label regions and fill definite counts first. Use complements to reduce counts when universal set known.

Solved example

Example: In 100 students, 60 like A, 50 like B, 30 like both. Then like at least one = 60+50−30=80.

Advanced techniques

Stars-and-bars for distributing identical objects: number of solutions to x1+...+xk = n (nonnegative integers) = C(n+k−1,k−1).
Use complementary counting to handle "none" or "at least one" easily.
In probability, linearity of expectation simplifies expected counts even when dependencies exist.
In permutations, arrange restricted objects first or use inclusion-exclusion for forbidden adjacencies.

50 Practice MCQs — Modern Math (Answers highlighted)

Questions include counting, distributions, conditional probability, Bayes, inclusion-exclusion, and expectation. Use the toggle above for quick/expanded view.

Q1. Number of ways to arrange letters of 'ABC' = ?

A. 3

B. 6

C. 9

D. 4

Q2. Choose 2 from 5: C(5,2) = ?

A. 10

B. 10

C. 5

D. 20

Q3. Number of ways to seat 5 people around a round table (rotations same) = ?

A. 120

B. 24

C. 20

D. 5

Q4. How many 4-digit numbers can be formed from digits 1-5 without repetition?

A. 625

B. 120

C. 5P4 = 120

D. 256

Q5. Number of ways to choose committee of 3 from 7 = ?

A. 35

B. 35

C. 21

D. 49

Q6. If probability of event A is 0.3, probability of its complement is?

A. 0.3

B. 0.7

C. 0.5

D. 1.3

Q7. Number of ways to arrange 'AAB' = ?

A. 3

B. 3 (3!/2!)

C. 6

D. 2

Q8. In flipping 2 fair coins, probability of exactly one head = ?

A. 1/2

B. 1/4

C. 3/4

D. 2/3

Q9. Number of subsets of a set with 4 elements = ?

A. 4

B. 16

C. 8

D. 2^4−1

Q10. If two dice rolled, probability sum = 7 = ?

A. 1/6

B. 1/6

C. 1/12

D. 1/3

Q11. Number of ways to put 3 identical balls into 4 boxes (allow empty) = ?

A. C(3+4−1,4−1)=C(6,3)=20

B. 20

C. 12

D. 15

Q12. If P(A)=0.5, P(B)=0.4 and A,B independent, P(A∩B)=?

A. 0.2

B. 0.9

C. 0.1

D. 0.4

Q13. Ways to choose 2 non-adjacent people from a circle of 6 = ?

A. 9

B. 9

C. 12

D. 6

Q14. Number of injective functions from a 3-element set to a 5-element set = ?

A. P(5,3)=60

B. 60

C. 125

D. 10

Q15. If 3 cards drawn without replacement from 52, probability all are hearts = ?

A. (13/52)^3

B. (13/52)×(12/51)×(11/50)

C. 1/4

D. 13/52

Q16. How many permutations of 4 letters from {A,B,C,D,E} allow A before B? (count orderings where A appears earlier than B)

A. 60/2=30

B. 30

C. 15

D. 24

Q17. If events A,B mutually exclusive, P(A∪B)=?

A. P(A)+P(B)

B. P(A)P(B)

C. P(A)−P(B)

D. None

Q18. Number of 5-letter strings from alphabet {A,B} = ?

A. 2^5 = 32

B. 10

C. 5

D. 25

Q19. Inclusion-exclusion: |A∪B∪C| = ?

A. |A|+|B|+|C| − |A∩B| − |B∩C| − |C∩A| + |A∩B∩C|

B. Correct formula above

C. Sum only

D. None

Q20. Expected value of fair die roll = ?

A. 3.5

B. 4

C. 3

D. 2.5

Q21. Number of ways to arrange 4 people in a row so that two specific people sit together = ?

A. 12

B. 12 (consider pair as block: 3!×2!)

C. 8

D. 6

Q22. If P(A|B)=P(A), events A and B are:

A. Independent

B. Mutually exclusive

C. Same

D. Complementary

Q23. Number of binary strings of length 6 with exactly three 1s = ?

A. C(6,3) = 20

B. 15

C. 10

D. 30

Q24. If sample space equally likely of size n, probability of event with k favourable outcomes = ?

A. k/n

B. n/k

C. k×n

D. 1/(k+n)

Q25. How many diagonals in a polygon with 10 sides? = ?

A. n(n−3)/2 → 10×7/2 = 35

B. 45

C. 40

D. 25

Q26. Probability of drawing ace from standard deck = ?

A. 4/52 = 1/13

B. 1/52

C. 4/13

D. 1/4

Q27. Number of onto functions from 2-element set to 2-element set = ?

A. 2

B. 2 (both bijections)

C. 4

D. 1

Q28. Ways to select 3 kings from a deck (there are 4 kings) = ?

A. C(4,3)=4

B. 6

C. 1

D. 12

Q29. Probability that a randomly chosen permutation of {1,2,3,4} has 1 before 2 = ?

A. 1/2

B. 1/4

C. 2/3

D. 1/3

Q30. Using stars and bars: number of solutions to x1+x2+x3=5 (nonnegative) = ?

A. C(5+3−1,3−1)=C(7,2)=21

B. 35

C. 15

D. 10

Q31. If events A,B independent and P(A)=0.6,P(B)=0.5, P(A∪B)=?

A. 0.8

B. 0.8 (0.6+0.5−0.3)

C. 0.9

D. 0.7

Q32. Number of ways to form a committee with at least one woman from 4 men and 3 women when committee size=3 = ?

A. Total − all men = C(7,3)−C(4,3)=35−4=31

B. 31

C. 30

D. 28

Q33. Probability of drawing two aces with replacement from deck = ?

A. (1/13)^2 = 1/169

B. 1/169

C. 1/52

D. 1/26

Q34. Number of ways to arrange 3 red and 2 blue balls in line = ?

A. 5!/(3!2!) = 10

B. 20

C. 30

D. 15

Q35. Bayes: Given disease prevalence 1% and test sensitivity 99% and false positive 5%, probability person with positive test actually has disease = ? (approx)

A. ~0.166

B. ≈ (0.01×0.99)/((0.01×0.99)+(0.99×0.05)) ≈ 0.166

C. 0.5

D. 0.01

Q36. Probability that two randomly chosen integers from 1..10 are coprime? (approx — quick)

A. ~0.6

B. 0.6 (approx — 0.6 is close)

C. 0.3

D. 0.9

Q37. Number of ways to choose a president and secretary from 10 people (order matters) = ?

A. P(10,2)=90

B. 90

C. 45

D. 100

Q38. If A∩B=∅, then |A∪B| = ?

A. |A|+|B|

B. |A|

C. |B|

D. 0

Q39. How many four-letter words (real or nonsense) can be made from 26 letters with repetition allowed? = ?

A. 26^4

B. 26P4

C. 26C4

D. 24^4

Q40. Expected number of heads in 10 fair coin tosses = ?

A. 5

B. 10

C. 2.5

D. 7

Q41. Number of surjective functions from a 3-element set to a 2-element set = ?

A. 6 (2^3 − 2 ways that miss an element) = 6

B. 8

C. 4

D. 2

Q42. Probability of drawing at least one ace in two draws with replacement = ?

A. 1 − (48/52)^2

B. (4/52)^2

C. 1/13

D. 2/13

Q43. Number of ways to arrange 8 distinct books on a shelf = ?

A. 8!

B. 64

C. P(8,8)

D. Both A and C

Q44. If |A|=20, |B|=15 and |A∩B|=5, |A∪B| = ?

A. 30

B. 25

C. 35

D. 20

Q45. Probability of drawing a red card from standard deck = ?

A. 1/2

B. 1/4

C. 26/52

D. Both A and C

Q46. Number of ways to choose committee of 4 from 10 if two specific people cannot both serve together = ?

A. C(10,4) − C(8,2) = 210 − 28 = 182

B. 182

C. 200

D. 1820

Q47. If expected value of X is E(X)=2 and E(Y)=3, what's E(3X+2Y)?

A. 3×2 + 2×3 = 12

B. 10

C. 6

D. 8

Q48. Number of ways to distribute 5 identical candies to 3 kids with each at least one = ?

A. C(5−1,3−1)=C(4,2)=6

B. 6

C. 10

D. 3

Q49. If P(A)=0.2,P(B)=0.3 and P(A∩B)=0.1, P(A|B)=?

A. 0.1/0.3 = 1/3

B. 0.2

C. 0.3

D. 0.5

Q50. Best general strategy for Modern Math in CAT:

A. Model with diagrams (trees/Venn/slots), use complement & linearity of expectation, and check symmetry

B. Memorize formulas only

C. Guess randomly

D. Avoid practice

Practice & Test-day Tips

Draw trees for sequential probability; label paths with probabilities and multiply along branches.
For counting, decide if order matters; choose permutations or combinations accordingly.
Use inclusion-exclusion for overlapping counts and complements for at-least-one events.
Practice expectation problems — they often simplify via linearity even when events dependent.