In how many ways can a set of two positive integers less than 100 be chosen?

Solution:

Combinations are selections made by taking some or all of a number of objects, irrespective of their arrangements.

The number of combinations of n different things taken r at a time, denoted by ⁿC_r and it is given by

ⁿC_r = n!/ [r! (n - r)!], where 0 ≤ r ≤ n.

Number of ways a set of two positive integers less than 100 be chosen

There are 99 integers we must choose set of two

So the number of ways can be written as

⁹⁹C₂ = 99!/ (97! × 2!) = (99 × 98)/2 = 99 × 49 = 4851

Therefore, a set of two positive integers less than 100 can be chosen in 4851 ways.

Example:

In how many ways can a set of two positive integers less than 50 be chosen?

Solution:

An integer is positive if it is greater than zero.

Example: 1, 2, 3 . . .

Number of ways a set of two positive integers less than 50 be chosen = 50/2 = 25

Therefore, a set of two positive integers less than 50 can be chosen in 25 ways.

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In how many ways can a set of two positive integers less than 100 be chosen?

Summary:

A set of two positive integers less than 100 can be chosen in 4851 ways.