‘Very small chance’ Queen’s ‘secret message’ is accident says number cruncher

There's a "very small chance" the Queen's secret "It's Coming Home" message to the Three Lions is an accident, says an intrepid Daily Star number cruncher.

Royal fans believe they have spotted the phrase in Her Majesty's good luck letter to the England team ahead of tonight's Euro 2020 final which has sent Twitter abuzz.

In her letter last night, the monarch praised the Three Lions for their "spirit, commitment, and pride" as she spoke of her memories of the 1966 World Cup when Captain Bobby Moore after the 4-2 win over Germany.

But royal fans looked a bit closer at the letter to find another message to Gareth Southgate and his squad.

One eagle-eyed Twitter user was able to spot the footie anthem "It's Coming Home" within her letter, written on Windsor Castle-headed paper, sharing a picture of the note with all the relevant letters circled.

So one has asked, what are the chances of that?

The Daily Star's fearless journalists gave the maths problem a go and came up with…. we're going to need a bigger calculator as it's just that small.

Buckingham Palace has declined to comment so here's the next best thing for our lucky readers.

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So you can work out the chances of something happening by dividing the number of ways it could happen by the number of different things that could happen at all.

For example, the chances of you getting 1 or 3 in a dice roll is 2 divided by 6, because there are two ways of it happening and six possible numbers that you can get in a roll.

As we can't read Her Majesty's mind the next best thing is to ask what are the chances of the message randomly popping up in a letter of the same length as Queen Elizabeth's.

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To do that, you'll need to count how long it is – 576 characters including spaces and punctuation – and also how many different letters you can write using the alphabet, numbers, punctuation and brackets as she's used in her letter.

If we're not bothered about whether the 'itscominghome' letters are in upper or lower cases, then with each character you have 47 possibilities: the 26 letters of the alphabet, 10 numbers, basic punctuation and the two brackets as well as the spaces between them.

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That done, all you need to do is count how many different ways you can get the letters of 'itscominghome' in order in a randomly written letter that's 576 characters long, where you can use any of 47 symbols.

Going back to the dice roll, working out the number of possibilities is the easy bit: you have 576 characters, and 47 options for each of them – so that's 47 options for the first character, and again for the second, and so on for the 576th one.

Mathematically, that mind-bogglingly big number comes to 47 to the power of 576, aka 47^576.

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Then the tricky bit – how many 'itscominghome' messages can you get in a letter?

What you can do is count it without insisting the letters are in order first: so 'i' has 576 options where it can appear in the letter, 't' then has 575 options and so on until 'e' which will be left with 563 spaces which haven't been taken by the other letters in 'its coming home'.

Then for each of these possible unordered secret messages, there can only be eight ways they can be in the correct order as there are two 'i', 'm's and 'o's in the phrase and every other letter appears once – that's two times two times two then, eight.

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So the proportion of the ones in order is 8/how many unordered ones there, which you get by counting again.

The first letter has 13 places to go, second 12 like above and so on. A shorthand for that number is 13! and so you have 8/13! as your proportion.

So the number of possible 'itscominghome' message comes to… 8/13! multiplied by 576*575*…*564.

And as that last number is the same as 576!/563!, hey presto – the probability is 8*576!/(563!*13!*47^576).

Which is a very small number, in the unlikely event that I haven't made a mistake, and you'll need a big calculator to see just how small.

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