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How many ways can you choose 10 mugs from a collection of 15?
(15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6) / (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2) =(15 x 14 x 13 x 12 x 11) / (5 x 4 x 3 x 2) =(14 x 13 x 12 x 11) / (8) = 3,003 different groupsof 10 mugs.
How do you change Roman numerals into numbers?
i only know the basics like I = 1 II = 2 III = 3 IV = 4 V =5 VI =6 VII = 7 VIII = 8Okay, this is a little complicated. It's looks as though there's a lot to learn from the length of my explanation, but I'm trying to make it as clear as possible, with lots of examples. Stay with me!Roman numbering is done by referring to certain fixed points on the number scale, and then either adding to them, or taking away. so let's start with the fixed points:I which means one (1)V which means five (5)X which means ten (10)L which means fifty (50)C which means one hundred (100)D which means five hundred (500)M which means one thousand (1000)After that you start putting them together so as to make the numbers in between.You can start with the very simplest form - add one number after another, this means that since I means one, if you add an extra I then you have one more.This means that, since I is one, V is 5 and X is 10:II = 2 (1 + 1)VI= 6 (5 + 1)XI - 11 (10 + 1)this works all the way up to MI, which is M plus I or 1001.You do the same thing to add a bit more, soIII = 3 (1 + 1 + 1)VII = 7 (5 + 1 + 1)XII = 12 (10 + 1 + 1)Try working out the way you would write 52, 102, 502 and 1002 for yourself!So far, so good. Now it gets a bit more complicated.Once you get a bit beyond the the things you can represent with three letters (like MIII), then having that many letters can get complicated, so the Romans did with their numbering system what many of us do with clocks. Officials may so "9.55", but most of us say "5 to 10" - in other words we start counting backwards.This is what the Romans did. Sometimes you will see 4 written as IIII, but mostly Roman numbers count back when you get to needing just one before something. This means that 4 is written as "one less than five". You've seen that VI (5 and 1) is six, with the I after the V to show it's more than V. Four is written as IV, with the I before the V to show that it is one less than five.This too applies as you go up the scale:IV = 4 (5 less 1)IX = 9 (10 less 1)IC = 99 (100 less 1)Again you can work out the rest yourself.What can get to you is that the "before" and "after" systems are used at the same time. This starts with the number 14.10 = X11 = XI12 = XII13 = XIIIso far, so good; all of these are ten plus one, ten plus two ones etc. When you get to 15 you don't add on five Is, you add on just one V - so 15 = XV. (10 + 5)But when you get to 14 you have ten plus "however you write 4" - which is IV. This means that14 = XIV (10 + one less than five!)15 = XV16 = XVI (10 + 5 + 1)17 = XVII (10 + 5 + 1 + 1)Once we get to twenty, we've reached another "double it" point - there is no special number for 20, so we write it as 2 tens "XX". Thirty is 3 tens XXX.So what happens to 19? Well, nineteen is one short of the second "X" in "XX", so we write it like that "XIX" (10 + 9, which we write as one less than ten)24 is two Xs and one less than V, so we write as XXIV. 29 is XXIX or XX (20) plus one less than X (IX).This "taking away" can be done with I, and also with X and C. This means that when you want to write 90, you remember it as "ten less than one hundred" - or XC. You also put the "less than" number just before the last of collection. This means that 199 becomes CIC, and 290 becomes CCXC (two hundred and "ten less than the final hundred").If you need to add on smaller letters, then you can do so. So 292 becomes CCXCII (one hundred, another hundred, ten less than a third hundred and one and one more).This is not quite as confusing as it seems, because Roman numbers do obey two rules.Firstly when you're adding numbers on, they always come in the same order - the biggest number first, next biggest after that, and so on.This means that MMXVII is 1000 plus 1000 (2000!), plus ten (X) plus V (five) plus I and I, giving you 2017.The second rule is that you can only use one letter for the "less than" rule. This means that when you see a letter out of order it must be on its own - so 28 is written as XXVIII (10 + 10 + 5 + 1 + 1 +1), whereas 29 is XXIX (10 + 10 + "one less than the last 10"). This is confusing to "decode", but after a while you just see "VII" as "seven" and "IX" as "nine" and "XC" as "ninety".All the same, it can still take a while before you have to work out MCMXCVIII.You take a quick look and see that there's a "C" out of order (before an "M") and then an "X" out of order before a "C". The rest of the letters are in the right order. This means that we have "One thousand" ("M", or 1000) plus "one hundred less than one thousand ("CM" or 900) plus ten less than one hundred ("XC" or 90) plus five plus one plus one.This sounds like a lot of adding, but take it in order - 1000 + 900 (1900) plus 90 (1990) plus five plus 3 ones, or eight, giving you 1998.Five, fifty and five hundred are always added on, and never doubled. So, you always write the "five" bit of a number in the right order - 15 is always XV, 35 is always thirty plus five (XXXV) and 1500 is always MD (1000 + 500). You add extra numbers from there - MDCVI is 1666.You can see why trading people took up Arabic numbers (1,2,3 ...10, 100,500 etc) when they reached Europe in the 12th and 13th centuries, not least because a longer number was always a bigger number. Adding II to MCMXCVIII gives you MM, which makes it impossible to add up numbers in columns.Still, before you conclude that the Romans were mugs, remember that they were famed for centuries for their engineering skills - and that takes maths. How they did that is another story.Please let me know if this answer helped you, as I spent a lot of time writing it up specially!
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