List of top Mathematics Questions on Permutation and Combination

If \( \alpha, \beta \) are roots of the equation \( 12x^2 - 20x + 3 = 0 \), \( \lambda \in \mathbb{R} \). If \( \frac{1}{2} \leq |\beta - \alpha| \leq \frac{3}{2} \), then the sum of all possible values of \( \lambda \) is:
In how many ways can the word ‘ESPECIALLY’ be arranged such that the vowels always come together?
What shall be the arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in 1421 including itself?
How many four digit numbers, which are divisible by 6, can be formed using the digits 0, 2, 3, 4, 6, such that no digit is used more than once and 0 does not occur in the left-most position?
There are 6 boxes numbered 1, 2, 3, 4, 5, 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is:
In how many ways can the letters of the word MATHEMATICS be arranged so that all the vowels always come together?
From a group of 7 men and 6 women, 5 persons are to be selected so that at least 3 men are on the committee. In how many ways can this be done?
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:
In a group of 3 girls and 4 boys, there are two boys \( B_1 \) and \( B_2 \). The number of ways in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but \( B_1 \) and \( B_2 \) are not adjacent to each other, is:

Let  R = {(1, 2), (2, 3), (3, 3)}} be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is:

The number of words that can be formed using all the letters of the word "DAUGHTER" such that all the vowels never come together, is:
There are 6 tasks and 6 persons. Task 1 cannot be assigned either to person 1 or person 2. Task 2 must be assigned to either person 3 or person 4. Every person is to be assigned one task. In how many ways can the assignment be done?