10. Six friends Pat, Qasim, Roman, Sam, Tara and Uma, stand in a line for a photograph. There are three people standing between Pat and Qasim, two between Qasim and Roman and one between Roman and Sam. Sam is not at either end of the line.
How many people are standing between Tara and Uma?
A 4
B 3
C 2
D 1
E 0
Solution
C
We indicate each of the friends by the first letter of their name, and a person whose name we are not yet sure about by an asterisk ( $*$ ).

We can assume, without loss of generality, that, from the point of view of the photographer, Qasim is to the right of Pat. Because there are three people standing between Pat and Qasim, the line is either
$$
* P * * * Q \text { or } P * * * Q * \text {. }
$$

There are two people between Qasim and Roman. Roman cannot be to the right of Quasim, because there is at most one friend to the right of Quasim. Therefore Roman is to the left of Quasim and the line is either
$$
* P R * * Q \text { or } P R * * Q * \text {. }
$$

There is one person between Roman and Sam. Therefore either Sam is immediately to the left of Pat, or immediately to the left of Qasim.
If Sam is immediately to the left of Pat, the line would be
$$
S P R * * Q \text {. }
$$

However this is impossible because Sam is not at either end of the line.
Therefore Sam is immediately to the left of Qasim. Hence the line is either
$$
* P R * S Q \text { or } P R * S Q *
$$
with Tara and Uma occupying the two places marked by asterisks.
We see that, however Tara and Uma occupy these two places, the number of people standing between Tara and Uma will be 2 .

For investigation
10.1 Specify one additional piece of information that would make it possible to work out the exact order of the six friends from left to right, as seen by the photographer.