The set of integers greater than or equal to 0 and less than 10, followed by an upper and lower case 'e'
Variance = sigma((value - mean)2) / (# values - 1) Mean = (0+1+1+2)/4 = 1 Variance = ((0-1)2+(1-1)2+(1-1)2+(2-1)2)/(4-1) Variance = (1+0+0+1)/3 Variance = 2/3 Variance ~ 0.667
The range is 6. (6 - 0 = 6)
I suggest you use an heuristic approach by setting yourself a computationally smaller problem, namely, how many combinations of 3 numbers in ascending order can be made from the digits 0 through 5? This is 'computationally smaller' in the sense that one can write down all of the combinations in a couple of minutes. (0, 1, 2) (0, 1, 3) (0, 1, 4) (0, 1, 5) (0, 2, 3) (0, 2, 4) (0, 2, 5) (0, 3, 4) (0, 3, 5) (0, 4, 5) (1, 2, 3) (1, 2, 4) (1, 2, 5) (1, 3, 4) (1, 3, 5) (1, 4, 5) (2, 3, 4) (2, 3, 5) (2, 4, 5) (3, 4, 5) Not only that, when I write them down in this way I see that, in writing them systematically (to ensure that I have them all) I notice that they are in ascending order! So this question is equivalent to asking how many ways there are of choosing 3 items from 6 (without any conditions). Going back to the original question I now know that one need not write out all of the combinations, I can simply calculate the number of ways of choosing 6 items from 52, which is 20358520.
11 + 4, 12 + 3, 13 + 2, 14 + 1, 15 + 0, 15 - 0, 16 - 1, 17 - 2, 18 - 3, 19 - 4, 20 - 5, 21 - 6, 22 - 7, 23 - 8, 24 - 9, 1*6 + 9, 1*7 + 8, 1*8 + 7, 1*9 + 6, 2*3 + 9, 2*4 + 7, 2*5 + 5, 2*6 + 3, 2*7 + 1, 2*8 - 1, 2*9 - 3 3*2 + 9, 3*3 + 6, 3*4 + 4, 3*5 + 0, 3*6 - 3, 3*7 - 6, 3*8 - 9 4*2 + 7, 4*3 + 3, 4*4 - 1, 4*5 - 5, 4*6 - 9 5*2 + 5, 5*3 + 0, 5*4 - 5 6*1 + 9, 6*2 + 3, 6*3 - 3, 6*4 - 9 7*1 + 8, 7*2 + 1, 7*3 - 6 8*1 + 7, 8*2 - 1, 8*3-9 9*1 + 6, 9*2 - 3 1*3*5 Is that enough?
There are 16 possible outcomes, each of which is equally likely. So each has a probability of 1/16 HHHH: X = H*T = 4*0 = 0 HHHT: X = H*T = 3*1 = 3 HHTH: X = H*T = 3*1 = 3 HTHH: X = H*T = 3*1 = 3 THHH: X = H*T = 3*1 = 3 HHTT: X = H*T = 2*2 = 4 HTHT: X = H*T = 2*2 = 4 HTTH: X = H*T = 2*2 = 4 THHT: X = H*T = 2*2 = 4 THTH: X = H*T = 2*2 = 4 TTHH: X = H*T = 2*2 = 4 HTTT: X = H*T = 1*3 = 3 THTT: X = H*T = 1*3 = 3 TTHT: X = H*T = 1*3 = 3 TTTH: X = H*T = 1*3 = 3 TTTT: X = H*T = 0*4 = 0 So, the probability distribution function of X is f(X = 0) = 1/16 f(X = 1) = 4/16 = 1/4 f(X = 2) = 6/16 = 3/8 f(X = 3) = 4/16 = 1/4 f(X = 4 = 1/16 and f(X = x) = 0 for all other x
I was only able to get 13 but I think there can be more, this is what i got: 0+0+4=4 4+0+0=4 0+4+0=4 0+1+3=4 0+3+1=4 0+2+2=4 1+2+1=4 1+1+2=4 1+3+0=4 2+2+0=4 2+0+2=4 3+1+0=4 3+0+1=4
+4 +3 +3 -4 -3 -2 0 -1 +3 +1+1 0 +1
2,2,0,5,1,4,1,3,0,0,1,4,4,0,1,4,3,4,2,1,0
8
each row has the contents of 8 5 3 in that order 8 0 0 5 0 3 5 3 0 2 3 3 2 5 1 7 0 1 7 1 0 4 1 3 4 4 0 the other solution is 8 0 0 3 5 0 3 2 3 6 2 0 6 0 2 1 5 2 1 4 3 4 4 0 See related link "Dividing" for more info
3. 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9 2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 1 1 7 0 6 7 9 8 2 1 4 8 0 8 6 5 1 3 2 8 2 3 0 6 6 4 7 0 9 3 8 4 4 6 0 9 5 5 0 5 8 2 2 3 1 7 2 5 3 5 9 4 0 8 1 2 8 4 8 1 1 1 7 4 5 0 2 8 4 1 0 2 7 0 1 9 3 8 5 2 1 1 0 5 5 5 9 6 4 4 6 2 2 9 4 8 9 5 4 9 3 0 3 8 1 9 6 4 4 2 8 8 1 0 9 7 5 6 6 5 9 3 3 4 4 6 1 2 8 4 7 5 6 4 8 2 3 3 7 8 6 7 8 3 1 6 5 2 7 1 2 0 1 9 0 9 1 4 5 6 4 8 5 6 6 9 2 3 4 6 0 3 4 8 6 1 0 4 5 4 3 2 6 6 4 8 2 1 3 3 9 3 6 0 7 2 6 0 2 4 9 1 4 1 2 7 3 7 2 4 5 8 7 0 0 6 6 0 6 3 1 5 5 8 8 1 7 4 8 8 1 5 2 0 9 2 0 9 6 2 8 2 9 2 5 4 0 9 1 7 1 5 3 6 4 3 6 7 8 9 2 5 9 0 3 6 0 0 1 1 3 3 0 5 3 0 5 4 8 8 2 0 4 6 6 5 2 1 3 8 4 1 4 6 9 5 1 9 4 1 5 1 1 6 0 9 4 3 3 0 5 7 2 7 0 3 6 5 7 5 9 5 9 1 9 5 3 0 9 2 1 8 6 1 1 7 3 8 1 9 3 2 6 1 1 7 9 3 1 0 5 1 1 8 5 4 8 0 7 4 4 6 2 3 7 9 9 6 2 7 4 9 5 6 7 3 5 1 8 8 5 7 5 2 7 2 4 8 9 1 2 2 7 9 3 8 1 8 3 0 1 1 9 4 9 1 2 9 8 3 3 6 7 3 3 6 2 4 4 0 6 5 6 6 4 3 0 8 6 0 2 1 3 9 4 9 4 6 3 9 5 2 2 4 7 3 7 1 9 0 7 0 2 1 7 9 8 6 0 9 4 3 7 0 2 7 7 0 5 3 9 2 1 7 1 7 6 2 9 3 1 7 6 7 5 2 3 8 4 6 7 4 8 1 8 4 6 7 6 6 9 4 0 5 1 3 2 0 0 0 5 6 8 1 2 7 1 4 5 2 6 3 5 6 0 8 2 7 7 8 5 7 7 1 3 4 2 7 5 7 7 8 9 6 0 9 1 7 3 6 3 7 1 7 8 7 2 1 4 6 8 4 4 0 9 0 1 2 2 4 9 5 3 4 3 0 1 4 6 5 4 9 5 8 5 3 7 1 0 5 0 7 9 2 2 7 9 6 8 9 2 5 8 9 2 3 5 4 2 0 1 9 9 5 6 1 1 2 1 2 9 0 2 1 9 6 0 8 6 4 0 3 4 4 1 8 1 5 9 8 1 3 6 2 9 7 7 4 7 7 1 3 0 9 9 6 0 5 1 8 7 0 7 2 1 1 3 4 9 9 9 9 9 9 8 3 7 2 9 7 8 0 4 9 9 5 1 0 5 9 7 3 1 7 3 2 8 1 6 0 9 6 3 1 8 5 9 5 0 2 4 4 5 9 4 5 5 3 4 6 9 0 8 3 0 2 6 4 2 5 2 2 3 0 8 2 5 3 3 4 4 6 8 5 0 3 5 2 6 1 9 3 1 1 8 8 1 7 1 0 1 0 0 0 3 1 3 7 8 3 8 7 5 2 8 8 6 5 8 7 5 3 3 2 0 8 3 8 1 4 2 0 6 1 7 1 7 7 6 6 9 1 4 7 3 0 3 5 9 8 2 5 3 4 9 0 4 2 8 7 5 5 4 6 8 7 3 1 1 5 9 5 6 2 8 6 3 8 8 2 3 5 3 7 8 7 5 9 3 7 5 1 9 5 7 7 8 1 8 5 7 7 8 0 5 3 2 1 7 1 2 2 6 8 0 6 6 1 3 0 0 1 9 2 7 8 7 6 6 1 1 1 9 5 9 0 9 2 1 6 4 2 0 1 9 8 9
1 (0+1) 3 (0+1+2) 6 (0+1+2+3) 10 (0+1+2+3+4) 15 (0+1+2+3+4+5) 21 (0+1+2+3+4+5+6) Notice these are the numbers you can arrange into equalateral triangles.
1 2 0 3 -1 4 -2 5 -3 6 or 1 2 3 4 5 6 ... interlaced with 0 -1 -2 -3 -4 -5 .....
The domain is {-1, 0, 2, 4}.
Points: (-2, -3) and (4, 0) Slope: (-3-0)/(-2-4) = 1/2
it will be 4 1+1=2-2=0+3=3-4=-1+5=4 regards, Aunali
e---------------4-2-0---------------------4-2-0-------------0-0------------------ B-0---------0----------------------0---0-0-----------4---2-4-------0------------- G---1----1-----------------1--3-----------------------------------------1-------- D------2---------------------------------------------------------------------2--- A-------------------------------------------------------------------------------- E-------------------------------------------------------------------------------- e---------------0--4-2-0---------------------4-2-0-------------0-0--------------- B-0---------0----------------------0---0-0-----------4---2-4-------0------------- G---1----1-----------------1--3-----------------------------------------1-------- D------2---------------------------------------------------------------------2--- A-------------------------------------------------------------------------------- E-------------------------------------------------------------------------------- e-4--4-4-5-7-7--5-4-2-4-5-5---5-4-2-0-----------0-------------------------------- B------------------------------------------------4--2-4------------0------------- G------------------------------------------------------------1--3---------------- D-------------------------------------------------------------------------------- A-------------------------------------------------------------------------------- E-------------------------------------------------------------------------------- e---------0-0--0-----------2--5-4------------------0-2-4-5-7--------------------- B-0--0----------4-2--2-2----------5--5--4--0-0----------------------------------- G-------------------------------------------------------------------------------- D-------------------------------------------------------------------------------- A-------------------------------------------------------------------------------- E-------------------------------------------------------------------------------- e-0--2--4---5---2---0------------------------------------------------------------- B--------------------------------------------------------------------------------- G--------------------------------------------------------------------------------- D--------------------------------------------------------------------------------- A--------------------------------------------------------------------------------- E---------------------------------------------------------------------------------