X+1/2(106+74)?
if x+1/2(106+74)=o than x= -90
What are some interesting things that equal 1?
Let's analyze the permutations of [math](1, 2, 3, ..., n)[/math]. Let's first do [math]n = 3[/math] for simplicity. [math](1, 2, 3), (1, 3, 2), (2, 1, 3),[/math] [math](2, 3, 1), (3, 1, 2), (3, 2, 1)[/math]. For each of these let's count how many position of the permutation coincide with the positions of the original permutation: [math](1, 2, 3). [/math]These are called fixed points. As a list,[math]3, 1, 1, 0, 0, 1[/math] for the [math]n = 3[/math] case. What is the average number of fixed points here? 1.Is this true for any [math]n[/math]? What is the average number of fixed points for a permutation of [math](1, 2, ..., n)[/math]? In fact, it's always 1 fixed point: and the proof is intuitive:Each position has a [math]\frac{1}{n}[/math] probability of being fixed. So over all [math]n[/math] positions that is [math]n \cdot \frac{1}{n} = 1[/math] fixed point! You can word this with probabilistic method or expected value also:Since precisely [math](n-1)![/math] permutations fix a specific point, the expected value of fixed points at any position is [math]\frac{(n-1)!}{n!} = \frac{1}{n}[/math] So let [math]X_i[/math] be the event that position [math]i[/math] is fixed. Then by linearity of expectation,[math] \mathbb{E}[X_1 + X_2 + ... + X_n] = [/math] [math]\mathbb{E}[X_1]+\mathbb{E}[X_2]+...+\mathbb{E}[X_n] = n \cdot \frac{1}{n} = 1 [/math]
How many QR codes can be generated?
More than the atoms in the universe (estimated to be 10x10x10... 81 times). More than all the email addresses created and yet to be created....It is a really really large number. The Wikipedia article mentions that it can have 7089 numeric only characters: ie equivalent to10x10x10..... 7089 times (each character can be 0-9).There is nothing magical about QR Codes. It is just machine speak and the storage dictates how much you can write. So how many different numbers can you write if you have a paper grid with 7089 squares ?Or if you have a sheet of paper in which you can type up-to 4296 alpha-numeric characters. How many different essays can you compose ? Mind-bogglingly large (if you want to calculate, it is 45x45x45 ....... 4296 times!)This picture illustrates the point. On the right is the QR Code machine speak version of the English text on the left.