explain n! and how to figure odds for lotteries?

Question

pick 6 with 51 #s is about 18000940 : 1, but pick 5 of 47, then 1 of 27 and the odds shoot up to 45million : 1

Answer 1

The mathematical symbol "!" means "factorial" (NOT "faculty," as one respondent suggested--how funny!!!). It is a shorthand way of indicating a special type of repeated multiplication.
1! = 1
2! = 2x1 = 2
3! = 3x2x1 = 6
4! = 4x3x2x1 = 24
So take the integer for which you need a factorial, then count down to 1 and multiply them all together.

Factorials are used in calculating things like lottery odds by using the idea of permutations and combinations. When you are dealing with lottery drawings, it should be a combination--not a permutation. The reason is because it doesn't matter which order you select the winning numbers, as long as you select them all. (In other words, the numbers "1,2,3" and "3,1,2" are the same in a lottery drawing--the order in which they were drawn does not affect the outcome.)

So the formula for calculating a combination is
n!/((n-r)!r!)
which means that the factorial in the top is the number of objects from which you are drawing. In the bottom will be two factorials--the number of objects you are drawing, and the difference between the number of objects and those being drawn. Let's do an example to help explain.

The first instance you mentioned was choosing 6 numbers out of 51. This is calculated as follows:
51! / ((51-6)!6!)
51! / (45! x 6!)

Now notice that 51! = 51x50x49x48x47x46x45!, wo I can cancel the 45! from the top and bottom, giving me

(51x50x49x48x47x46) / (6x5x4x3x2x1)
18,009,460
This is slightly different from the figure you reported.

Let's look at your other example. You want to draw 5 numbers out of 47. This would be calculated as
47! / ((47-5)!5!)
47! / (42!x5!)
(47x46x45x44x43) / (5x4x3x2x1)
1,533,939

In addition, you need to select 1 out of 27. There are 27 possibilities here. If you need to calculate it, 27!/((27-1)!1!) = 27!/(26!x1!) = 27.

So I need the 5 out of 47 AND the 1 out of 27, which means multiply those results.
1533939 x 27 = 41,416,353

The question you are asking is not just about the math, though. Since the second example has fewer numbers to select from--and since you are ultimately drawing 6 balls either way--shouldn't it be easier to get a winning number when there are fewer balls involved? Why is it that the second scheme has higher odds? That is what you are really asking.

Notice the key difference in the way it is calculated, though. In the first scheme, the sixth number is involved in the calculation of the combination--so there are six values on the top, and there is a 6 in the denominator. Look at the fraction again:
(51x50x49x48x47x46) / (6x5x4x3x2x1)
In the second scheme you have a combination involving five values, and then the result is multiplied by another value. Here is that calculation:
[(47x46x45x44x43) / (5x4x3x2x1)] x 27
So in the first one, the "last ball drawn" adds the 46 in the numerator and the 6 in the denominator--which means we are adding a factor of almost 8 (46/6 is 7 2/3). In the second one, however, we are adding a factor of 27--there is no "offsetting" value added to the denominator when we select the last ball.

So the difference between them--the reason that the odds are worse in the second scenario you presented--is because of the fact that the last ball in the second scenario is drawn separatly. That changes the way the odds of that value get multiplied into the problem, which really increases the odds.

Answer 2

An example of n!:

If n =7 then 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
You just multilply n by all the natural numbers less than n.

It is relatively easy to compute lottery odds. The odds of having the winning ticket when choosing 6 out of 51 = 180009460:1. You missed the 6, but you obviously used the correct formula (or calculator) which is:

nCr = n! / ((n-r)!*r!) with n = 51 and r = 6

For 5 out of 47, your odds are 1533939:1
For 1 out of 27 = 27:1 (I think you meant 1 out of 47)
If you multiply these two together you get:

27 x 1533939 = 41,416,353:1
47 x 1533939 = 72,095,133:1

I think what confused you was that you missed the 6 and so you thought it was 18 million instead 180 million, and that doesn't make sense.

One other calulation that comes in handy, when other prizes besides the jackpot are awarded, is the odds of getting some of your numbers correct. Here's a couple of fun/useful calculators:

http://www.csgnetwork.com/oddscalc.html

http://www.lottogenie.com/html/odds.html

Here's a link that provides info on tons of real lotteries and gives odds tables and also the formulas that were used to compute the odds:

http://www.johnph77.com/math/stlot.html#toc

PS Good Luck!

Answer 3

Lottery Probability Formula

Answer 4

How To Calculate Lottery Odds

Answer 5

Pick 6 with 51 numbers.

(51x50x49x48x47x46) divided by (1x2x3x4x5x6)

Pick 5 of 47 and then 1 of 27

47x46x45x44x43 divided by 1x2x3x4x5 . Multiply the result by 27.

Answer 6

from what i've learn this yr which is in the counting technic's chapter, n! is pronouns as 'n factorial' whcih represent
n x (n-1) x (n-2) x ...........x 3 x 2 x 1
eg. 9! = 9x8x7x6x5x4x3x2x1 = 362880

if u hv a graphic clc. it wud be even easier.. b/c they're in the program..

as in the lottery.. it's impossible to find the probability.. b/c there were too much number... and thus the probability of say no 32 to out is 1/1000.. [tht's estimately]

Answer 7

I can explain n! (pronounced "n faculty"). It basically says that you have to multiply the number with each whole number less than itself. Eg.

n = 5
n! = 5*4*3*2*1 = 120

Answer 8

what is your question?