JEE Advanced (One or More Correct Option): Permutations & Combinations | Chapter-wise Tests for JEE Main & Advanced PDF Download

Q.1. The number of ways in which 1- candidates A₁, A₂,….,A₁₀ can be ranked, so that A₁ is always above A₂ is
(a) 10! / 2
(b) 8! x ¹⁰C₂
(c) ¹⁰P₂ 
(d) ¹⁰C₂

Correct Answer is Options (a, b)
The number of ways of placing A₁ and A₂ in ten places so that A₁ is always above A₂ is ¹⁰C₂. There are 8! ways of arranging the eight other candidates.
Hence, total number of arrangements
=  

Q.2. If n > 1, then (1 + x)ⁿ - nx - 1 is divisible by
(a) x
(b) x² 
(c) x³ 
(d) x⁴

Correct Answer is Options (a, b)
We have, (1 + x)ⁿ - nx - 1 = C₀ + C₁x + C₂x² ... + C_nxⁿ - nx -1
=  
Thus, (1 + x)ⁿ - nx - 1 is divisible by x² and hence by x

Q.3. If then k may lie in the region
(a) (∞ , 2)
(b) [-2, √3 )
(c) [-√3, 0]
(d) (√3, 2]

Correct Answer is Options (b, d)

Q.4. There are n married couples in a party. Each person shakes hand with every person other than her or his spouse. The total number of handshakes must be
(a) ²ⁿC₂ -n
(b) ²ⁿC₂ - ( n- 1)

(c) 2n( n - 1)
(d) ²ⁿC₂

Correct Answer is Options (a, c)
Total handshakes possible = ²ⁿC₂
This will include n handshakes in which a person shakes hand with her or his spouse.
⇒ Required number = ²ⁿC₂ – n = 2n(n – 1)

Q.5. The number of selections of four letters taken from the word COLLEGE must be
(a) 22
(b) 18
(c) coefficient of x4 in the expansion of (1 + x + x²)² (1 + x)³
(d) coefficient of x4 in the expansion of (1 + x + x²)³ (1 + x)²

Correct Answer is Options (b, c)
We have LL; EE; C, O, G Þ Number of selections of four letters = coefficient of x⁴ in (x⁰ + x¹ + x²)(x⁰ + x¹ + x²)(x⁰ + x¹)(x⁰ + x¹)(x⁰ + x¹)
= coefficient of x⁴ in (1 + x + x²)²(1 + x)³
⇒ (C) is correct. Since coefficient can be found as 18, (B) is also correct.

Q.6. If  then
(a) r = 7
(b) r = 8
(c) r = 9
(d) r = 10

Correct Answer is Options (a, b, c, d)

⇒
So r = 7, 8, 9, 10 are possible answers.

Q.7. Triangles are formed by joining vertices of a octagon then number of triangle
(a)  In which exactly one side common with the side of octagon is 32
(b)  In which atmost one side common with the side of polygon is 48
(c) At least one side common with the side polygon 50
(d)  Total number of triangle 56

Correct Answer is Options (a, b, d)
Total number of triangle = ⁸C₃ = 56
Number of triangle having exactly one side common with the polygon = 8 × 4 = 32
Number of triangle having exactly two side common with the polygon = 8
Number of triangle having no side common with the polygon = 16

Q.8. Total number of ways in which four boys and four girls can be seated around a round table, so that no two girls sit together, is equal to
(a) 4!5!
(b) 3! 4!
(c) 5(4!)² 
(d) 4(3!)²

Correct Answer is Options (b, d)
The boys can be arrange in 3! ways around the circle. They form 4 gaps in which girls will occupy difference gaps. So, the girls can now be arrange in 4! ways. So, the total number of ways = 3! 4! = 4 × (3!)²

Q.9. The position vector of a point P is  when x, y, z ∈ N and  

the number of possible position of P is  
(a) 36
(b) 72
(c)  66
(d) ⁹C₂

Correct Answer is Options (a, d)

The required number of positions
=^10-1C_3-1 =  ⁹C2  = 36

Q.10. The number of non-negative integral solution of x₁ + x₂ + x₃ + x₄ ≤ n (where n is a positive integer) is
(a) ⁿ⁺³C₃
(b) ⁿ⁺⁴C₄
(c) ⁿ⁺⁵C₅
(d) ⁿ⁺⁴C_n

Correct Answer is Options (b, d)
Let x5 be such that x₁ + x₂ + x₃ + x₄ + x₅ = n, we now seek the non-negative integral solutions of x1 + x2 + x3 + x4 + x5 = n. The number of required solutions = ⁿ⁺⁴C₄ = ⁿ⁺⁴C_n