1. Oleg has 1000 rubles, and he wants to give his mother tulips for March 8, with the condition that the number of tulips must be odd, and no color shade should be repeated. In the store where Oleg went, one tulip costs 49 rubles, and there are twenty shades of flowers available. How many ways can Oleg give his mother the flowers?
Solution. From the condition, it is obvious that the maximum number of flowers in a bouquet is 20.

1st method

Using the property of binomial coefficients

$$
\mathrm{C}_{n}^{1}+\mathrm{C}_{n}^{3}+\mathrm{C}_{n}^{5}+\cdots=2^{n-1}
$$

and also considering their combinatorial meaning, we get that the number of ways to form a bouquet with an odd number of flowers of no more than 20 different shades (with the condition that no shade should be repeated) is:

$$
C_{20}^{1}+C_{20}^{3}+C_{20}^{5}+\cdots+C_{20}^{19}=2^{19}
$$

2nd method

Consider 19 flowers of 19 different shades. The number of ways to form a bouquet from these flowers without considering the parity is $2^{19}$. If the bouquet has an odd number of flowers, we keep it; if it has an even number, we add the unused twentieth flower. Thus, the total number of ways to form a bouquet is $2^{19}$.

## Answer: $2^{19}$.