Question

There are 62 phones in an office building.

How many unique connections between two of these phones can be made?

Answer 1

To find the number of unique connections between two phones, we can use the concept of combinations. Since a connection between two phones does not depend on the order in which the phones are chosen, we need to calculate the number of ways to choose 2 phones from 62 phones.

The formula for combinations \( C(n, r) \) is given by:

\[
C(n, r) = \frac{n!}{r!(n - r)!}
\]

where
- \( n \) is the total number of items,
- \( r \) is the number of items to choose, and
- \( ! \) denotes factorial, the product of all positive integers up to that number.

In this case, \( n = 62 \) (the number of phones), and \( r = 2 \) (the number of phones we want to connect).

Using the formula:

\[
C(62, 2) = \frac{62!}{2!(62 - 2)!} = \frac{62!}{2! \cdot 60!}
\]

This simplifies as follows:

\[
= \frac{62 \times 61 \times 60!}{2! \times 60!}
\]

The \( 60! \) in the numerator and denominator cancels out:

\[
= \frac{62 \times 61}{2!}
\]

Calculating \( 2! \):

\[
2! = 2 \times 1 = 2
\]

Now substituting that back into our equation:

\[
= \frac{62 \times 61}{2} = \frac{3782}{2} = 1891
\]

Thus, the number of unique connections between two of the 62 phones is \( \boxed{1891} \).