# CAT Question Bank - Ordered & Unordered Solutions

• Q20) How many different triangles are there with integer sides and with perimeter = 2009?

• Q21) Number of ways of distributing 10 distinct balls in 3 identical boxes?

• Q22) In how many ways can 14 identical toys be distributed among 3 boys, such that each boy gets at least one toy and no two boys get the same number of toys?

• Q23) How many ordered pairs of integers (a, b), are there such that their product is a positive integer less than 100? Also, if (a,b) is not different from (b,a), then how many such pairs are possible?

• Q24) In how many ways can 83 be expressed as the sum of two natural numbers that are coprime to each other?

• Q25) Find the number of ordered triplets (a, b, c ) such that b = a + 2 and c = ab - 2 and a, b , and c are prime numbers.
a) only one
b) only two
c) infinitely many
d) none

• Q26) Find the number of ordered pairs (x, y) where both x and y are non-negative integers such that, x – (1/y) = (x/y) + 1?

• Q27) Find the number of ordered prime triplets (a, b, c) such that a, b and c are distinct and ab + bc + ca = 316.
a. 0
b. 6
c. 10
d. Infinitely many

• Q28) How many unordered triplets of positive integers p, q, r exist such that p/q + q/r + r/p = 2?
a. 0
b. 1
c. 2
d. More than 2

• Q29) How many ways can 200 be written as Sum of 3 even natural numbers

• Q30) a, b, c are three distinct integers from 2 to 10 (both inclusive). Exactly one of ab, bc and ca is odd. abc is a multiple of 4. The arithmetic mean of a and b is an integer and so is the arithmetic mean of a, b and c. How many such triplets are possible (unordered triplets)?
a. 8
b. 6
c. 2
d. 4

• @rowdy-rathore ordered =9
unordered =9

• @rowdy-rathore 0 as triplets cannot be formed

• @Kushal-Khandelwal

Funda : a/x + b/y = 1/k where a, b and k are positive integers and we want number of value of (x ,y) satisfying this equation
First find number of factors of a * b * k^2
let number of factors = M
a) total number of positive integral solutions = M
b) total integral solutions = 2 * M - 1

143^2 = 11^2 x 13^2
Number of factors = 3 * 3 = 9
Total number of positive integral solutions = 9
Total integral solutions = 2 * 9 - 1 = 17

• @Kushal-Khandelwal

Funda: Number of ordered pairs possible for LCM of N = P1^a x P2^b x P3^c
= (2a + 1) (2b + 1) (2c + 1)

Here 360 = 2^3 * 3^2 * 5
So number of ordered pairs possible = (2 * 3 + 1) (2 * 2 + 1)(2 * 1 + 1) = 105

More on this concept can be read @ Formula To Find Ordered & Unordered Pairs Possible For A Given LCM - Hemant Malhotra

• @Kushal-Khandelwal

Funda: Number of Triangles with Integer sides for a given perimeter.

• If the perimeter p is even then, total triangles is [p^2]/48.
• If the perimeter p is odd then, total triangles is [(p+3)^2]/48
• If it asks for number of scalene triangle with a given perimeter P, then subtract 6 and apply the same formula . For even [(p-6)^2]/48 and for odd [(p-3)^2]/48.

Here P = 2009 (odd)
So total triangles possible = (2012)^2/48 = 84336

• @Kushal-Khandelwal

AM ≥ GM
(p/q + q/r + r/p)/3 ≥ 1
(p/q + q/r + r/p) ≥ 3
So minimum value possible is 3.
No triplets exist.
[ @shashank_prabhu ]

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