. * In how many different ways can 5 identical coins be placed in 3 different boxes?
We draw the pieces as $\mathbf{O}$ and the edges of the boxes as $\mathbf{I}$. There are exactly $\binom{7}{2}=21$ ways to draw a sequence of symbols $\mathbf{O}$ and $\mathbf{I}$ containing two $\mathbf{I}$.

Note: since there are only three boxes, we can also manage by counting directly.

## Proof by contradiction: some examples

Proof by contradiction involves assuming that a mathematical proposition we wish to prove is false, and hoping to derive a contradiction from this assumption. We have given two examples, which are very classic, of proof by contradiction. These are also explained (much better than here) in the photocopy by Xavier from the Saint-Malo workshop in 2003.

As an example, we have proven the mathematical proposition: There are infinitely many prime numbers.

The negation of the proposition to be proven is that the set of prime numbers is finite. Suppose this is the case, and let $\{2,3,5,7, \ldots 113, \ldots p\}$ be this set (with $p$ being the largest prime number). Then let $\mathrm{P}=1+2 \times 3 \times 5 \times 7 \times \cdots \times 113 \times \cdots \times p$. $\mathrm{P}$ cannot be a prime number (it is strictly greater than $p$); but then, if $q$ is its smallest divisor different from 1, $q$ must necessarily be in the list of prime numbers we have written. This is absurd, because $P-1$ is also a multiple of $q$; it would therefore have to divide the number 1.