The glass container is shown filled to a height of 2.25 inches. How much sand is currently in the container.

Mathematics

1 answer:

JulijaS [17]3 years ago

3 0

2.25 inches of sand.

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Let n be a positive integer and define [n] to be the set of the first n positive integers. That is, [n] = {1, 2, 3, . . . , n}.

yaroslaw [1]

Answer: There are $2^{n-1}$ ways of doing this

Hi!

To solve this problem we can think in term of binary numbers. Let's start with an example:

n=5, A = {1, 2 ,3}, B = {4,5}

We can think of A as 11100, number 1 meaning "this element is in A" and number 0 meaning "this element is not in A"

And we can think of B as 00011.

Thinking like this, the empty set is 00000, and [n] =11111 (this is the case A=empty set, B=[n])

This representation is a 5 digit binary number. There are $2^5$ of these numbers. Each one of this is a possible selection of A and B. But there are repetitions: 11100 is the same selection as 00011. So we have to divide by two. The total number of ways of selecting A and B is the $2^{5-1} = 2^4$ .