Questions covered in Hard

Review the question prompts below before starting the timed test. Try each problem independently, then use the interactive test to record your answers and review the worked solutions.

1. The number of ways in which the letters of the word 'ARRANGE' can be arranged such that the two R's are never together is
2. If $^nC_{12} = ^nC_8$, then $^nC_{17}$ is equal to
3. The number of rectangles that can be formed on a chessboard is
4. The sum of the digits in the unit place of all the numbers formed with the digits 1, 2, 3, 4, 5 taken all at a time is
5. Find the number of different signals that can be generated by arranging at least 2 flags in order on a vertical staff, if 5 different flags are available.
6. How many numbers greater than 1,000,000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?
7. In how many ways can 5 girls and 3 boys be seated in a row so that no two boys are together?
8. Find $r$ if $5 \cdot ^4P_r = 6 \cdot ^5P_{r-1}$.
9. A committee of 5 is to be formed out of 6 men and 4 women. In how many ways can this be done when (i) at least 2 women are included? (ii) at most 2 women are included?
10. Find the rank of the word 'ZENITH' in a dictionary.
11. From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?
12. Case Study: Using the digits 0, 1, 2, 3, 4, 5.
    (i) How many 4-digit numbers can be formed if repetition is allowed?
    (ii) How many 4-digit numbers divisible by 5 can be formed if repetition is allowed?
    (iii) How many 4-digit numbers strictly greater than 3000 can be formed if repetition is not allowed?