C3. All the telephone numbers in Georgetown have six digits and each of them begins with the digits 81 . Kate finds the scrap of paper shown, with part of Jenny's telephone number on it.
How many different possibilities are there for Jenny's telephone number?
Solution
The scrap of paper could read ' 1018 ' or ' 8101 '.
Let us start by working out which parts of the phone number Kate could have been seeing. We show the scrap of paper as a rectangle.
$$
81018 d
$$

There are ten possibilities for the unknown digit $d$.
$$
811018
$$

There is only one possibility.
$$
8101 c d
$$

There are one hundred possibilities for the pair of unknown digits $c$ and $d$.
$$
818101
$$

There is only one possibility.
However, all the possibilities in the first case-' $81018 d$ '-are also counted under the third case-' $8101 c d$ '.

Hence there are in fact 102 possibilities for Jenny's telephone number.