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NMTC Combinatorics Problems and Solutions

NMTC 2010 Primary Stage 1 Question 1

$\mathrm{n}, \mathrm{a}$ are natural numbers each greater than 1 . If $a+a+a+a+\ldots+a=2010$, and there are $n$ terms on the left hand side, then the number of ordered pairs $(a, n)$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

14

NMTC 2019 Primary Stage 1 Question 10

Sum of the odd numbers from 1 to 2019 both inclusive, is divisible by

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

both 100 and 101

NMTC 2019 Primary Stage 1 Question 4

$X$ is a 5 digit number. Let $Y$ be the sum of the digits of $X$. Let $Z$ be the sum of the digits of $Y$. Then the maximum possible value that $Z$ can have is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

12

NMTC 2019 Primary Stage 1 Question 6

Look at the set of numbers ${2,3,5,7,8,10,12}$. Four numbers are selected from this and made into two pairs. The pairs are added and the resulting two numbers are multiplied. The smallest such product is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

60

NMTC 2019 Primary Stage 1 Question 12

A calendar for 2019 is made using 4 sheets, each sheet having 3 months. The total number of days shown in each of the four sheets $\left(1^{\text {st }}, 2^{\text {nd }}, 3^{\text {red }}, 4^{\text {th }}\right)$ respectively is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

(90,91,92,92)

NMTC 2019 Primary Stage 1 Question 13

Triples of odd numbers $(a, b, c)$ with $a<b<c$, with $a, b, c$ from 1 to 10 are generated such that $a+b+c$ is prime number. The number of such triple is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

6

NMTC 2019 Primary Stage 1 Question 17

Given below is the triangular form of AMTI.

$\mathrm{A}$
A M A
A M TM A
A M T I T M A


The number of ways you can spell AMTI, top to bottom, right to left or left to right or a combination of these is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

15

NMTC 2019 Primary Stage 1 Question 25

A string of beads has a recurring pattern as follows: 5 blue, 4 black, 4 white, 5 blue, 4 black, 4 white ……….. and so on. The colour of the $321^{\text {st }}$ bead is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

Black

NMTC 2019 Sub Junior Stage 1 Question 6

In a $5 \times 5$ grid having 25 cells, Janani has to enter 0 or 1 in each cell such that each sub square grid of size $2 \times 2$ has exactly three equal numbers. What is the maximum possible sum of the numbers in all the 25 cells put together?

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

21

NMTC 2019 Sub Junior Stage 1Question 20

Numbers of 5 -digit multiples of 13 is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

6923

NMTC 2019 Sub Junior Stage 1 Question 27

In a room, $50 \%$ of the people are wearing gloves, and $80 \%$ of the people are wearing hats. The minimum percentage of people in the room wearing both a hat and a glove is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

30

NMTC 2019 Junior Stage 1 Question 12

Given a sheet of 16 stamps as shown, the number of ways of choosing three connected stamps (two adjacent stamps must have an edge in common) is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

42

NMTC 2019 Junior Stage 1 Question 18

A $4 \times 4$ anti-magic square is an arrangement of the numbers 1 to 16 in a square so that the totals of each of the four rows, four columns and the two diagonals are ten consecutive numbers in some order. The diagram shows an incomplete anti magic square. When it is completed, the number in the position of ${ }^{*}$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

16

NMTC 2019 Junior Stage 1 Question 20

In a stack of coins, each row has exactly one coin less than the row below. If we have nine coins, two such towers are possible. Of these, the tower on the left is the tallest. If you have 2015 coins, the height of the tallest towers is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

62

NMTC 2019 Junior Stage 1 Question 23

In a single move a King $\mathrm{K}$ is allowed to move to any of the squares touching the square it is on, including diagonals, as indicated in the figure. The number of different paths using exactly seven moves to go from $A$ to $B$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

127

NMTC 2010 Primary Stage 1 Question 1

$\mathrm{n}, \mathrm{a}$ are natural numbers each greater than 1 . If $a+a+a+a+\ldots+a=2010$, and there are $n$ terms on the left hand side, then the number of ordered pairs $(a, n)$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

14

NMTC 2019 Primary Stage 1 Question 10

Sum of the odd numbers from 1 to 2019 both inclusive, is divisible by

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

both 100 and 101

NMTC 2019 Primary Stage 1 Question 4

$X$ is a 5 digit number. Let $Y$ be the sum of the digits of $X$. Let $Z$ be the sum of the digits of $Y$. Then the maximum possible value that $Z$ can have is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

12

NMTC 2019 Primary Stage 1 Question 6

Look at the set of numbers ${2,3,5,7,8,10,12}$. Four numbers are selected from this and made into two pairs. The pairs are added and the resulting two numbers are multiplied. The smallest such product is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

60

NMTC 2019 Primary Stage 1 Question 12

A calendar for 2019 is made using 4 sheets, each sheet having 3 months. The total number of days shown in each of the four sheets $\left(1^{\text {st }}, 2^{\text {nd }}, 3^{\text {red }}, 4^{\text {th }}\right)$ respectively is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

(90,91,92,92)

NMTC 2019 Primary Stage 1 Question 13

Triples of odd numbers $(a, b, c)$ with $a<b<c$, with $a, b, c$ from 1 to 10 are generated such that $a+b+c$ is prime number. The number of such triple is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

6

NMTC 2019 Primary Stage 1 Question 17

Given below is the triangular form of AMTI.

$\mathrm{A}$
A M A
A M TM A
A M T I T M A


The number of ways you can spell AMTI, top to bottom, right to left or left to right or a combination of these is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

15

NMTC 2019 Primary Stage 1 Question 25

A string of beads has a recurring pattern as follows: 5 blue, 4 black, 4 white, 5 blue, 4 black, 4 white ……….. and so on. The colour of the $321^{\text {st }}$ bead is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

Black

NMTC 2019 Sub Junior Stage 1 Question 6

In a $5 \times 5$ grid having 25 cells, Janani has to enter 0 or 1 in each cell such that each sub square grid of size $2 \times 2$ has exactly three equal numbers. What is the maximum possible sum of the numbers in all the 25 cells put together?

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

21

NMTC 2019 Sub Junior Stage 1Question 20

Numbers of 5 -digit multiples of 13 is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

6923

NMTC 2019 Sub Junior Stage 1 Question 27

In a room, $50 \%$ of the people are wearing gloves, and $80 \%$ of the people are wearing hats. The minimum percentage of people in the room wearing both a hat and a glove is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

30

NMTC 2019 Junior Stage 1 Question 12

Given a sheet of 16 stamps as shown, the number of ways of choosing three connected stamps (two adjacent stamps must have an edge in common) is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

42

NMTC 2019 Junior Stage 1 Question 18

A $4 \times 4$ anti-magic square is an arrangement of the numbers 1 to 16 in a square so that the totals of each of the four rows, four columns and the two diagonals are ten consecutive numbers in some order. The diagram shows an incomplete anti magic square. When it is completed, the number in the position of ${ }^{*}$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

16

NMTC 2019 Junior Stage 1 Question 20

In a stack of coins, each row has exactly one coin less than the row below. If we have nine coins, two such towers are possible. Of these, the tower on the left is the tallest. If you have 2015 coins, the height of the tallest towers is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

62

NMTC 2019 Junior Stage 1 Question 23

In a single move a King $\mathrm{K}$ is allowed to move to any of the squares touching the square it is on, including diagonals, as indicated in the figure. The number of different paths using exactly seven moves to go from $A$ to $B$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

127

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