Permutations and Combinations

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is .

If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at 440th position in this arrangement is :

1. PRNAKU
2. PRKAUN
3. PRKANU
4. PRNAUK

Let nCr-1 = 28, nCr = 56 and nCr+1 = 70. Let A(4 t, 4 t), B(2 t, -2 t) and C (3r - n, r2 - n - 1 ) be the vertices of a triangle ABC, where t is a parameter. If (3x - 1)2 + (3y)2 = , is the locus of the centroid of triangle ABC, then equals

1. 18
2. 8
3. 20
4. 6

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is

1. 5720
2. 5719
3. 4608
4. 4607

The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is .

Let S=\p1,p2 ,p10\ be the set of first ten prime numbers. Let A=S P, where P is the set of all possible products of distinct elements of S. Then the number of all ordered pairs (x,y), x S, y A, such that x divides y, is .

The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is .

Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to :

1. 8925
2. 9100
3. 8575
4. 8750

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is :

1. 34000
2. 37000
3. 35000
4. 36000

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is .