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Subject

Mathematics

Chapter

Ch 4Combinatorics and Mathematical Induction

Weightage~7marksapprox

4.1IntroductionRead →

4.2Fundamental principles of countingRead →

4.3Factorials16 Q

4.4Permutations20 Q

4.5Combinations25 Q

4.6Mathematical induction40 Q

CBSE Sample Papers33 Q

Q1The number of diagonals that can be drawn by joining the vertices of an octagon is: (a) 20 (b) 28 (c) 24 (d) 48 Q2How many numbers divisible by 5 and lying between 5000 and 6000 can be formed from the digits 5, 6, 7, 8 and 9 ?Q3How many different signals can be made by hoisting 6 differently coloured flags one above the other, when any number of them may be hoisted at one time ?Q4(a) Prove by mathematical induction that 7²ⁿ + 16n - 1 is divisible by 64 for all n ∈ N. **OR** (b) Using binomial theorem, find the value of 11⁷.Q5The number of five digit numbers in which all digits are even, is: (a) 4× 5⁴ (b) 4× 5⁵ (c) 5⁵ (d) 5× 5 Q6Write the relationship between Permutation and Combination.Q7Count the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digits are repeated.Q8(a) Prove that for any natural number n, aⁿ-bⁿ is divisible by a-b, where a>b. **OR** (b) Evaluate: ∫ dfrac2x+4x²+4x+6dx Q9There are 15 points in a plane and 5 of them are collinear. The number of straight lines joining any two points is: (a) 45 (b) 86 (c) 76 (d) 96 Q10Find the number of ways of arranging the letters of the word 'BANANA'.Q11If (n+2)C₈ : (n-2)P₄ = 57 : 16, find n.Q12(a) By the principle of Mathematical Induction, prove that for n ≥ 1, 1·2 + 2·3 + 3·4 + … n(n+1) = dfracn(n+1)(n+2)3. **OR** (b) In a triangle ABC, prove that dfracasin(B-C)b²-c² = dfracbsin(C-A)c²-a² = dfraccsin(A-B)a²-b².Q13The number of 5 digit numbers all digits of which are odd is: (a) 5⁶ (b) 25 (c) 625 (d) 5⁵ Q14Find the distinct permutations of the letters of the word ACCESSIBILITY.Q15(a) By the principle of mathematical induction, prove that, for n ≥ 1, 1³+2³+3³+....+n³ = (dfracn(n+1)2)². **OR** (b) Find dfracdydx, if x=a(t-sin t), y=a(1-cos t).Q16There are 8 points in a plane and 4 of them are collinear. The number of straight lines joining any 2 points is: (a) 39 (b) 45 (c) 38 (d) 23 Q17Number of sides of a polygon having 44 diagonals is: (a) 11 (b) 4 (c) 22 (d) 4!Q18If n ∈ N, then 7²ⁿ + 3³ⁿ⁻³· 3ⁿ⁻¹ is always divisible by: (a) 45 (b) 25 (c) 55 (d) 35 Q19If ⁿ⁻¹C₃ + ⁿ⁻¹C₄ > ⁿC₃ then: (a) n > 7 (b) n > 5 (c) n > 4 (d) n > 6 Q20If ⁿC₄ = 495, find the value of n.Q21If ⁿCr-1 = 36, ⁿCr = 84 and ⁿCr+1 = 126 then find the value of r.Q22There are n locks and n matching keys. If all the locks and keys are to be perfectly matched, then the maximum number of trials is: (a) n(n-1) (b) n(n+1) (c) n (d) dfracn(n+1)2 Q23ⁿC₀+ⁿC₁+......+ⁿCₙ= (a) 2ⁿ⁺¹ (b) 2ⁿ (c) 2ⁿ⁻¹ (d) 2n Q24If dfrac17!+dfrac18!=dfracA9! then find the value of A.Q25Find the number of ways of arranging the letters of the word INDIA.Q26If ⁿPr=720 and ⁿCr=120 find n, r.Q27The number of 5 digit numbers, all digits of which are odd is: (a) 5⁶ (b) 25 (c) 625 (d) 5⁵ Q28What is the unit digit of the sum 3! + 4! + … + 20!?Q29If (n+2)P₄ = 42 × ⁿP₂, find n.Q30(a) By the principle of mathematical induction, prove that, for n ≥ 1 : 1³+2³+3³+…+n³ = (dfracn(n+1)2)² **OR** (b) Prove that 1+a 1 1 \1 1+b 1 \1 1 1+c = abc(1+dfrac1a+dfrac1b+dfrac1c)Q31Number of sides of a polygon having 44 diagonals is: (a) 11 (b) 4 (c) 22 (d) 4!Q32If ⁿC₁₂ = ⁿC₉, find ²¹Cₙ.Q33Prove that dfrac(2n)!n! = 2ⁿ(1.3.5…(2n-1))

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Ch 4 — Combinatorics and Mathematical Induction

Question 114 of 134

Q. The number of 5 digit numbers all digits of which are odd is:

(a)

5^6

(b) 25

(c) 625

(d)

5^5

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