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UGC NET PAPER 1 · UNIT 05 · MODULE 6: GEOMETRY, COUNTING AND PROBABILITY · TOPIC 15

Mensuration, Permutation-Combination and Probability (Basics)

Lower-priority but formula-based: areas and volumes, nPr and nCr and basic probability.

  • 6 sub-topics
  • 6 min read
  • Topic 15 of 17

What this topic covers

6 sub-topics, in the order to study them. Tap one to jump to it.

  1. §OverviewThe official syllabus ends with "etc." after the named arithmetic topics, so these three areas can appear occasionally.
  2. 15.1Plane figuresFigure Area Perimeter or other Rectangle (l, b) l x b 2(l + b); diagonal = square root of (l squared + b squared) Square (side a) a squared 4a; diagonal = a x square root of…
  3. 15.2SolidsSolid Volume Surface area Cube (edge a) a cubed total 6a squared; diagonal a x square root of 3 Cuboid (l, b, h) l x b x h total 2(lb + bh + hl); diagonal = square root of…
    • Scaling rule
    • Polygon angles
  4. 15.3Mensuration examplesCovers 1540, 15 pi, 38,808.
    • 1540
    • 15 pi
    • 12 pi
    • 38,808
    • 5,544
  5. 15.4Permutations and combinationsCovers 34,650.
    • Examples
    • 34,650
  6. 15.5ProbabilityCovers 7/8, 5/36, 7/13, 5/14.
    • Standard sample spaces to know
    • Examples
    • 7/8
    • 5/36
    • 7/13
  7. 15.6TrapsUsing permutation when order does not matter, or combination when it does. Forgetting the complement shortcut for "at least one" problems. Forgetting to subtract the overlap…

Revision mode hides the full explanations and keeps each section's key terms, lists and tables. Use it from your second read onward.

§Overview#

The official syllabus ends with "etc." after the named arithmetic topics, so these three areas can appear occasionally. Treat them as lower priority than topics 8 to 14, but learn the formulas below because the questions are short and formula-based.

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15.1Plane figures#

In brief

Figure Area Perimeter or other Rectangle (l, b) l x b 2(l + b); diagonal = square root of (l squared + b squared) Square (side a) a squared 4a; diagonal = a x square root of…

Figure Area Perimeter or other
Rectangle (l, b) l x b 2(l + b); diagonal = square root of (l squared + b squared)
Square (side a) a squared 4a; diagonal = a x square root of 2
Triangle (base b, height h) 1/2 x b x h Heron: square root of [s(s - a)(s - b)(s - c)], s = (a + b + c)/2
Equilateral triangle (side a) (square root of 3/4) x a squared 3a
Right triangle 1/2 x base x height hypotenuse squared = base squared + height squared
Parallelogram base x height 2(sum of adjacent sides)
Rhombus 1/2 x d1 x d2 4 x side
Trapezium 1/2 x (sum of parallel sides) x height sum of all sides
Circle (radius r) pi x r squared circumference 2 x pi x r
Sector (angle theta) (theta/360) x pi x r squared arc = (theta/360) x 2 x pi x r
Semicircle 1/2 x pi x r squared pi x r + 2r

Use pi = 22/7 when the radius or diameter is a multiple of 7, otherwise use 3.14 or 3.1416 as the question says.

Pythagorean triples to memorise: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (9, 40, 41), and their multiples (6, 8, 10), (9, 12, 15), (10, 24, 26).

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15.2Solids#

In brief

Solid Volume Surface area Cube (edge a) a cubed total 6a squared; diagonal a x square root of 3 Cuboid (l, b, h) l x b x h total 2(lb + bh + hl); diagonal = square root of…

  • Scaling rule
  • Polygon angles
Solid Volume Surface area
Cube (edge a) a cubed total 6a squared; diagonal a x square root of 3
Cuboid (l, b, h) l x b x h total 2(lb + bh + hl); diagonal = square root of (l squared + b squared + h squared)
Cylinder (r, h) pi x r squared x h curved 2 x pi x r x h; total 2 x pi x r (r + h)
Cone (r, h, slant l) 1/3 x pi x r squared x h curved pi x r x l; total pi x r (r + l); l = square root of (r squared + h squared)
Sphere (r) 4/3 x pi x r cubed 4 x pi x r squared
Hemisphere (r) 2/3 x pi x r cubed curved 2 x pi x r squared; total 3 x pi x r squared

Scaling rule: if every length of a figure is multiplied by k, areas are multiplied by k squared and volumes by k cubed. Doubling the radius of a circle makes the area 4 times as large; doubling the radius of a sphere makes the volume 8 times as large.

Polygon angles: sum of interior angles of an n-sided polygon = (n - 2) x 180 degrees. Each interior angle of a regular polygon = (n - 2) x 180/n degrees. The sum of exterior angles is 360 degrees.

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15.3Mensuration examples#

In brief

Covers 1540, 15 pi, 38,808.

  • 1540
  • 15 pi
  • 12 pi
  • 38,808
  • 5,544

  • A rectangle of 12 by 5 has diagonal = square root of (144 + 25) = 13.

  • A cylinder with r = 7 and h = 10 has volume = 22/7 x 49 x 10 = 1540.

  • A cone with r = 3 and h = 4 has slant height 5, curved surface = pi x 3 x 5 = 15 pi, and volume = 1/3 x pi x 9 x 4 = 12 pi.

  • A sphere of radius 21 has volume = 4/3 x 22/7 x 9261 = 38,808 and surface area = 4 x 22/7 x 441 = 5,544.

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15.4Permutations and combinations#

In brief

Covers 34,650.

  • Examples
  • 34,650
Concept Formula or rule
Fundamental principle "And" (one task after another) multiplies the ways; "or" (alternative tasks) adds the ways
Factorial n! = n x (n - 1) x ... x 1, with 0! = 1
Permutation (order matters) nPr = n!/(n - r)!
Combination (order does not matter) nCr = n!/[r!(n - r)!]
Symmetry and identities nCr = nC(n - r); nC0 = nCn = 1; nCr + nC(r - 1) = (n + 1)Cr
Arrangements with repeated items n!/(p! x q! x r!) for p, q, r identical items
Circular arrangements (n - 1)! (or (n - 1)!/2 for a necklace or garland)
Items that must stay together Treat them as one block, then multiply by the arrangements inside the block
Handshakes among n people nC2
Diagonals of an n-sided polygon n(n - 3)/2

Examples

  • Number of 3-digit numbers from the digits 1 to 5 without repetition = 5P3 = 60.

  • Ways to choose 3 men from 5 and 2 women from 4 = 5C3 x 4C2 = 10 x 6 = 60.

  • Arrangements of the letters of LEVEL = 5!/(2! x 2!) = 30.

  • Arrangements of the letters of MISSISSIPPI = 11!/(4! x 4! x 2!) = 34,650.

  • 5 people in a row with two particular people always together: 4! x 2! = 48.

  • Handshakes among 10 people = 10C2 = 45. Diagonals of a hexagon = 6 x 3/2 = 9.

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15.5Probability#

In brief

Covers 7/8, 5/36, 7/13, 5/14.

  • Standard sample spaces to know
  • Examples
  • 7/8
  • 5/36
  • 7/13
  • 5/14
Concept Formula or rule
Probability of an event P(E) = favourable outcomes/total outcomes, always between 0 and 1
Complement P(not E) = 1 - P(E)
Addition rule P(A or B) = P(A) + P(B) - P(A and B)
Mutually exclusive events P(A or B) = P(A) + P(B)
Independent events P(A and B) = P(A) x P(B)
Conditional probability P(A given B) = P(A and B)/P(B)
Odds in favour of E Favourable : unfavourable

Standard sample spaces to know

  • One coin has 2 outcomes; n coins have 2 to the power n. One die has 6 outcomes; two dice have 36.

  • A deck has 52 cards: 4 suits of 13, 26 red and 26 black, 12 face cards (J, Q, K), and 4 aces.

  • Two dice sum to 7 in 6 ways (probability 1/6) and to 12 in 1 way (probability 1/36).

Examples

  • At least one head in 3 tosses = 1 - 1/8 = 7/8.

  • Sum of 8 on two dice = (2,6), (3,5), (4,4), (5,3), (6,2) = 5/36.

  • A red card or a king = 26/52 + 4/52 - 2/52 = 7/13.

  • A bag has 5 red and 3 blue balls, and 2 are drawn without replacement. Both are red with probability 5C2/8C2 = 10/28 = 5/14.

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15.6Traps#

In brief

Using permutation when order does not matter, or combination when it does. Forgetting the complement shortcut for "at least one" problems. Forgetting to subtract the overlap…

  • Using permutation when order does not matter, or combination when it does.

  • Forgetting the complement shortcut for "at least one" problems.

  • Forgetting to subtract the overlap in addition-rule problems (a red king counted twice).

  • Using 22/7 when the radius is not a multiple of 7.

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↑ Module 6: Geometry, Counting and Probability · All 18 topics in Mathematical Reasoning · Unit overview and sample MCQs

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