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The number of different words which can be formed from the letters of the word LUCKNOW when the vowels always occupy even places is

(a) 120 (b) 720 (c) 400 (d) None of these Solution: Total number of letters = 7 Number of vowels = 2 There are 3 even... View Article

Six teachers and six students have to sit around a circular table such that there is a teacher between any two students. The number of ways in which they can sit is

(a) 6 ! × 6 ! (b) 5 ! × 6 ! (c) 5 ! × 5 ! (d) None of these Solution: Six students can be arranged around... View Article

The greatest common divisor of ²⁰C₁, ²⁰C₃, …²⁰C₉ is

(a) 20 (b) 4 (c) 5 (d) None of these Solution: 20C1 = 20 20C3 = 20 × 19 × 3 = 1140 20C5 = 20C15 = 19... View Article

The expression ⁿC_r + 4.ⁿC_r-1 + 6.ⁿC_{r – 2} + 4.ⁿC_{r – 3} + ⁿC_{r – 4} is equal to

(a) n + 4Cr (b) 2.n + 4Cr – 1 (c) 4.nCr (d) 11.nCr Solution: nCr + nCr-1 = n!/r!(n-r)!] + [n!/(r-1)!(n-r+1)!] =... View Article

Total number of words that can be formed using all letters of the word FAILURE which neither begin with F nor end with E is equal to

(a) 3720 (b) 5040 (c) 3600 (d) 3480 Solution: The total number of words formed without any restriction = 7! The... View Article

A shopkeeper sells three varieties of perfumes and he has a large number of bottles of the same size of each variety in this stock. There are 5 places in a row in his showcase. The number of different ways of displaying the three varieties of perfumes in the showcase is

(a) 6 (b) 50 (c) 150 (d) None of these Solution: Number of ways of displaying without restriction = 35 Number of ways in... View Article

The number of permutations of the letters of the word HINDUSTAN such that neither the pattern ‘HIN’ nor ‘DUS’ nor ‘TAN’ appears are

(a) 166674 (b) 169194 (c) 166680 (d) 181434 Solution: There are 9 letters in HINDUSTAN in which N is repeated twice.... View Article

Find the number of non negative solutions of the system of equations: a + b = 10, a + b + c + d = 21, a + b + c + d + e + f = 33, a + b + c + d + e + f + g + h = 46, and so on till a + b + c + d + … + x + y + z = 208.

(a) 22P10 (b) 22P11 (c) 23P13 (d) None of these Solution: Consider the equation a + b = 10 number of solutions of this equation is... View Article

Find the number of non negative integral solutions of equation a + b + c + d = 20.

(a) 1800 (b) 1771 (c) 1770 (d) none of the above Solution: Number of non-negative integral solutions of the type of equations x1 + x2 +... View Article

Find the number of non-negative integral solutions of the equation x + y + z = 10.

(a) 10C2 (b) 12C2 (c) 11C2 (d) 9C2 Solution: Number of non-negative integral solutions of the type of equations x1 + x2 + … + xn... View Article

There are 18 points in a plane such that no three of them are in the same line except five points which are collinear. The number of triangles formed by these points is :

(a) 816 (b) 806 (c) 805 (d) 813 Solution: We need three non collinear points to form a triangle. The number of... View Article

The total number of 3-digit numbers, the sum of whose digits is even, is equal to

(a) 450 (b) 350 (c) 250 (d) 325 Solution: Let the three-digit number be n = pqr Sum of digits = p+q+r Since p+q+r... View Article

The number of ways in which a mixed doubles game in tennis can be arranged from 5 married couples, if no husband and wife play in the same game, is

(a) 46 (b) 54 (c) 60 (d) None of these Solution: Given that there are 5 married couples. Let the sides of the game... View Article

Let 1 ≤ m < n ≤ p. The number of subsets of the set A = {1, 2, 3, ……..p} having m, n as the least and the greatest elements respectively, is

(a) 2n-m-1-1 (b) 2n-m-1 (c) 2n-m (d) 2p-n+m-1 Solution: Given 1 ≤ m < n ≤ p A = {1, 2, 3, ........p} n is the... View Article

n’ is selected from the set {1, 2, 3, …, 100} and the number 2ⁿ + 3ⁿ + 5ⁿ is formed. Total number of ways of selecting ‘n’ so that the formed number is divisible by 4, is equal to

(a) 50 (b) 49 (c) 48 (d) None of these Solution: If n is odd, 3n = 4λ1 - 1 5n = 4λ2 + 1 => 2n... View Article

If 12 persons are seated in a row, the number of ways of selecting 3 persons from them, so that no two of them are seated next to each other is

(a) 85 (b) 100 (c) 120 (d) 240 Solution: The number of ways of selecting 3 persons from 12 people under the given... View Article

All the words that can be formed using alphabets A, H, L, U, and R are written as in a dictionary (no alphabet is repeated). The rank of the word RAHUL is

(a) 71 (b) 72 (c) 73 (d) 74 Solution: No. of words starting with A are 4 ! = 24 No. of words starting with H are 4... View Article

A boat is to be manned by eight men of whom 2 can only row on the bow side and 1 can only row on stroke side, the number of ways in which the crew can be arranged is

(a) 4360 (b) 5760 (c) 5930 (d) None of these Solution: 2 men on the bow side can be arranged in 4P2 ways = 12 1 man on... View Article

A person invites a party of 10 friends at dinner and place them so that 4 are on one round table and 6 on the other round table. The number of ways in which he can arrange the guests is

(a) 10!/6! (b) 10!/24 (c) 9!/24 (d) none of these Solution: 6 friends are at one round table and the remaining 4 are at the other round... View Article

A student performed the experiment to measure the speed of sound in air using the resonance air-column method. Two resonances in the air column were obtained by lowering the water level. The resonance with the shorter air-column is the first resonance and that with the longer air-column is the second resonance. Then,

a) the intensity of the sound heard at the first resonance was more than that at the resonance b) the prongs of the tuning fork were kept in a... View Article