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IMPORTANT
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There are n balls in a box, and the balls are numbered 1, 2, 3, …, n respectively. One of the balls is removed from the box, and it turns out that the sum of the numbers on the remaining balls in the box is 5048 . If the number on the ball removed from the box is m, find the value of m.

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Important Questions on Counting

HARD
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 27
Consider the 800 -digit integer 234523452345⋯2345. The first m digits and the last n digits of the above integer are crossed out so that the sum of the remaining digits is 2345 . Find the value of m+n10.
MEDIUM
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IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 28
The total number of integers between 0 and 105 having the digit sum equal to 8 is n, then find 5500-n.
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IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 29
How many ordered pairs of integers m,n, where 0<m<n<2008 satisfy the equation 20082+m2=20072+n2?
MEDIUM
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 30
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?
EASY
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 31
Mr. and Mrs. Yadav want to name their baby Yadav so that its monogram (first, middle, and last initials) will be in alphabetical order with no letters repeated. If total number of possible monograms are X, find X30.
MEDIUM
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 32
Nine chairs in a row are to be occupied by six students and professors Alpha. Beta and Gamma. These three profressors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can professors Alpha, Beta and Gamma choose their chairs?
HARD
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 33
A list of 2018 positive integers has a unique mode, which occurs exactly 10 times. Then the least number of distinct values that can occur in the list is N. Find the value of N5.
HARD
IOQM - PRMO and RMO
IMPORTANT
Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 34
Let n be a positive integer greater than 4 such that the decimal representation of n! ends in k zeros and the decimal representation of 2n ! ends in 3k zeros. Let s denote the sum of the four least possible values of n. What is the sum of the digits of s?