Let p be a prime and m be a positive integer. By mathematical induction on m, or otherwise, prove that whenever r is an integer such that p does not divide r, p divides
My Self Assessment
The letters of the word COCHIN are permuted and all permutations are arranged in an alphabetical order as in the English dictionary. The number of words that appear before the word COCHIN is
a) 360
b) 192
c) 96
d) 48
Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. In how many different ways can we place the balls so that no box is empty?
m men and n women are to be seated in a row so that no two women sit together. If m > n, then find the number of ways in which they can be seated.
The relatives of a man comprise 4 ladies and 3 gentlemen and his wife has 7 relatives 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that so that three of man’s relatives and three of wife’s relatives are included?
A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can three balls be drawn from a box if at least one black ball is to be included in the draw?
A student is allowed to select at most n books from a collection of (2n + 1) books. If the total number of ways in which he can select at least one book is 63, find the value of n?
Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to be on a particular side and three others on the other side. Determine the number of ways in which the seating arrangements can be made?
For a fixed value of n D = Then show that is divisible by n
A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be if at least five women have to be in the committee? In how many ways in these committees (i) The women are in majority, (ii)The men are in majority
The sides AB, BC and CA of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using these interior points as vertices is . . . .
Total number of ways in which six ‘+’ and four ‘’ signs can be arranged in a line so that no two ‘’signs occur together is …..
The ones and tens digits in 1! + 2! + 3! + . . .+ 49! are
a) 4 and 1
b) 3 and 1
c) 4 and 2
d) 5 and 1
If S = (1) (1!) + (2) (2!) + (3) (3!) + . . . + n (n!) then
a)
b)
c)
d) None of the above
n – 1Pr + r . n – 1Pr – 1 is equal to
a) n + 1Pr
b) nPr
c) r . nPr
d) None of these
The positive integer r such that 15C3r = 15Cr + 3 is equal to
a) 3
b) 4
c) 5
15C8 + 15C9 − 15C6 − 15C7 is equal to
a) 8
b) 0
c) 6
If 3 . x + 1C2 + 2P2 . x = 4 . xP2, x ∈ N, then x is equal to
a) 2
c) 3
The number of positive terms in the sequence is
b) 3
c) 4
The value of is
a) 28 – 2
b) 28 – 1
c) 28 + 1
d) 28
The least positive integral value of x which satisfies the inequality 10Cx – 1 > 2 . 10Cx , is
a) 7
b) 8
c) 9
d) 10
The value of when both numerator and denominator have their greatest values is
d)
The number of ways in which n different prizes can be distributed among m (<n) persons if each is entitled to receive at most n−1 prizes is
a) nm − n
b) mn
c) mn – m
The number of numbers greater than 50000 that can be formed by using the digits 3, 5, 6, 6, 7 is
a) 36
b) 48
c) 54
