[ Rule of product ] [ Combinations and permutations ]

How many different five-color flags with five vertical stripes of the same width can be made if eleven colors of fabric can be used? (Here, a flag is considered simply a piece of cloth, not attached to a pole or anything else.)

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See problems $\underline{30325}$ and $\underline{60374}$.

## $[$ Case Analysis $]$ Problem 78204 Topics: [ Compositions and Partitions $]$  [ Parity ]  Difficulty: $3-$  Grades: 8,9

List all monetary amounts, expressed in whole numbers of rubles, that can be represented both by an even and an odd number of banknotes. (In circulation, there were banknotes with denominations of $1, 3, 5, 10$, 25, 50, and 100 rubles.)

## Solution

First, assume that we have a monetary amount of 10 rubles or more. We will show that any such amount can be represented both by an even and an odd number of banknotes. Specifically, any even monetary amount (10 rubles or more) can be represented, on the one hand, by an even number of 1-ruble banknotes, and on the other hand, by one 10-ruble banknote and an even number of 1-ruble banknotes. Similarly, any odd monetary amount (10 rubles or more) can be represented, on the one hand, by an odd number of 1-ruble banknotes, and on the other hand, by one 10-ruble banknote and an odd number of 1-ruble banknotes.

Now consider monetary amounts less than 10 rubles. For representing these amounts, we only have banknotes with denominations of $1, 3, 5$ rubles—all odd. It is clear that an odd number of "odd" banknotes can only form an odd sum; therefore, no even amount less than 10 rubles can be represented by an odd number of banknotes. Similarly, an even number of "odd" banknotes can only form an even sum; therefore, no odd amount less than 10 rubles can be represented by an even number of banknotes.

## Answer

All amounts not less than 10 rubles.