## Task 2 - 030922

In a shooting competition, a shooter scored 40 points with 5 shots on a ten-ring target. With each shot, he hit at least 7 rings.

How many possibilities are there for the rings scored with each individual shot?

Note: The order must be taken into account. For example, 7, 7, 7, 9, 10 and 7, 7, 7, 10, 9 are considered different possibilities.
Without considering the order, there are the following possibilities:

1. $7,7,7,9,10 \quad ;$
2. $7,7,8,8,10$
3. $7,7,8,9,9 \quad ;$
4. $7,8,8,8,9$
5. $8,8,8,8,8$

and only these.

Observing the order leads to the calculation of the number of so-called arrangements (permutations) with repetition.

1. If all 5 numbers were different, there would be $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ different possibilities for arrangement. Since, however, the 7 appears three times, $3 \cdot 2 \cdot 1$ possibilities collapse into one, so that $\frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{3 \cdot 2 \cdot 1}=20$ possibilities remain.
2. and 3. analogous considerations lead to $\frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{2 \cdot 1 \cdot 2 \cdot 1}=30$ possibilities.
3. as 1.
4. 1 possibility.

In total, there are $20+30+30+20+1=101$ possibilities.