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IMPORTANT

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$m$ and $n$ are two positive integers satisfying $1 \leq m \leq n \leq 40 .$ Find the number of pairs of $(m, n)$ such that their product $m n$ is divisible by $33$ .

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Important Questions on Counting

EASY

IOQM - PRMO and RMO

IMPORTANT

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

HARD

IOQM - PRMO and RMO

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?

HARD

IOQM - PRMO and RMO

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

\frac{N - 1}{5} .

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 16

The numbers

1, 2, 3, 4, 5

are to be arranged in a circle. An arrangement is bad if it is not true that for every

n

from

1

15

one can find a subset of the numbers that appear consecutively on the circle that sum to

n .

Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 17

A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 18

What is the sum of the last two digits of the integer

1! + 2! + 3! + \dots + 2005! ?

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 20

There are

N

three digit numbers, lying between

100

and

999

inclusive, have two and only two consecutive digits identical. Then find the value of

\frac{N}{9} .

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 3 - Counting>EXERCISE>Q 21

A set of

16

square blocks is arranged into a

4 \times 4

square. If the number of combinations of

3

blocks that can be selected from that set so that no two are in the same row or column is

m

, then

\frac{m}{4}

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.

Important Questions on Counting

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$m$ and $n$ are two positive integers satisfying $1 \leq m \leq n \leq 40 .$ Find the number of pairs of $(m, n)$ such that their product $m n$ is divisible by $33$ .