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Five students A, B, C, D and E form a team to take part in a 5-leg relay competition. If A cannot run the first leg and D cannot run the last leg, how many ways can we arrange them to run the relay?

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Important Questions on Counting

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IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 8
Different positive 3-digit integers are formed from the five digits 1, 2, 3, 5, 7, and repetitions of the digits are allowed. As an example, such positive 3-digit integers include 352, 577, 111, etc. Now, the sum of all the distinct positive 3-digit integers formed in this way is written as ABC50. Find A+B+C.
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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 9

As shown in the picture, the knight can move to any of the indicated squares of the 8×8 chessboard in 1 move. If the knight starts from the position shown, find the number of possible landing positions after 20 consecutive moves.

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HARD
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IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 10
There are N number of 11-digit positive integers such that the digits from left to right are non-decreasing. (For example, 12345678999, 55555555555, 23345557889. Find the sum of all the digits of N.
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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 11
A four digit number consists of two distinct pairs of repeated digits (for example 2211, 2626 and 7007). Find the total number of such possible numbers that are divisible by 7 or 101 but not both.
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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 12
m and n are two positive integers satisfying 1≤m≤n≤40. Find the number of pairs of m,n such that their product mn is divisible by 33.
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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 13
How many 7 digit palindromes (numbers that read the same backward as forward) can be formed using the digits 2, 2, 3, 3, 5, 5, 5?
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IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 14
Ajay refuses to sit next to either Bharat or Chandan. Deepak refuses to sit next to Enees. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?
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IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 15
If there are N integers between 100 and 999, inclusive, have the property that some permutation of its digits is a multiple of 11 between 100 and 999? For example, both 121 and 231 have this property. Then find the value of N-15.