CAT Question Bank - Ordered & Unordered Solutions
Rowdy Rathore last edited by Rowdy Rathore
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
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.
d. Infinitely many
Q28) How many unordered triplets of positive integers p, q, r exist such that p/q + q/r + r/p = 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)?
@rowdy-rathore ordered =9
@rowdy-rathore 0 as triplets cannot be formed
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
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
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
zabeer last edited by zabeer
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 ]