Combinatorics | Number Theory Answers

i) Kishan is a sales executive who has to make visits to Colombo, Kandy, Galle,

Gampaha, Panadura and Negombo. He has to visit each city once only and always

takes the unique shortest route between any two cities.

a. How many possible routes are there in total for his journey? (3 marks)

b. Kishan has been told he has to start his journey from Colombo. In this

scenario how many possible routes are there? (2 marks)

c. IfKishan was to pick a route for his entire journey for all six cities at

random, what is the probability that the route he selected would start from

Colombo? (3 marks)

d. Kishan's manager calls him and tells him there is not enough time to visit

every city and to choose only four. How many possible routes are there for

Kishan to take in this scenario?

Identify whether each of the followings is permutation or combination and find the

answers.

(i) lsha wants to invest 12 million in three projects. She decided to invest 4 million in

each project and if seven shortlisted projects are available, in how many possible

ways she can invest her money?

(ii) Instead of equal allocation, she decided to invest 3million, 4 million and 5 million for

each project. How many possible arrangements are available for her investment

decision?

The formula for calculating the sum of all natural integers from 1 to n is well-known:

Sn = 1 + 2 + 3 + ... + n =

n

2 + n

2

Similary, we know about the formula for calculating the sum of the first n squares:

Qn = 1 · 1 + 2 · 2 + 3 · 3 + ... + n · n =

n

3

3

+

n

2

2

+

n

6

Now, we reduce one of the two multipliers of each product by one to get the following sum:

Mn = 0 · 1 + 1 · 2 + 2 · 3 + 3 · 4 + ... + (n − 1) · n

Find an explicit formula for calculating the sum Mn.

The formula for calculating the sum of all natural integers from 1 to n is well-known:

Sn = 1 + 2 + 3 + ... + n =

n

2 + n

2

Similary, we know about the formula for calculating the sum of the first n squares:

Qn = 1 · 1 + 2 · 2 + 3 · 3 + ... + n · n =

n

3

3

+

n

2

2

+

n

6

Now, we reduce one of the two multipliers of each product by one to get the following sum:

Mn = 0 · 1 + 1 · 2 + 2 · 3 + 3 · 4 + ... + (n − 1) · n

Find an explicit formula for calculating the sum Mn.

Find the smallest positive integer N that satisfies all of the following conditions:

• N is a square.

• N is a cube.

• N is an odd number.

• N is divisible by twelve prime numbers.

How many digits does this number N have?

The formula for calculating the sum of all natural integers from 1 to n is well-known:

Sn = 1 + 2 + 3 + ... + n =

n

2 + n

2

Similary, we know about the formula for calculating the sum of the first n squares:

Qn = 1 · 1 + 2 · 2 + 3 · 3 + ... + n · n =

n

3

3

+

n

2

2

+

n

6

Now, we reduce one of the two multipliers of each product by one to get the following sum:

Mn = 0 · 1 + 1 · 2 + 2 · 3 + 3 · 4 + ... + (n − 1) · n

Find an explicit formula for calculating the sum Mn.

hannah is making a game board that has a dimension of 15 inches by 24 inches if she will use square tiles what is the largest tile she can use

For each pair of numbers find integers 𝑥 and 𝑦 such that 𝑎𝑥 + 𝑏𝑦 =gcd (𝑎,𝑏)

a) 𝑎= 91, 𝑏=10

 A code have 4 digits in a specific order, the digits are between 0-9. How many different permutations are there if one digit may only be used once?

Find a counterexample to the statement “every positive integer greater than 7 can be written as the sum of the squares of three (not necessarily unique) integers.”