# How Many Keyboard Shortcuts Are There

Have you ever pressed key combination like
`Ctrl`+`Alt`+`⌦ Delete`?
or
`F10` `e` `c`?
(in Microsoft Windows, that activates menu, edit, and copy.)

How many chord combinations are there? Let's find out.

If we consider all the usual modifiers: `Alt` `Ctrl` `Shift` and `❖ Window`
Key, a total of 4 modifiers (not consider the right side modifiers for now). If we can use only 1 modifier, we have

`Alt``Ctrl``Shift``❖ Window`

we write:

# 4 things take 1, 4 possibilities C[4,1] = 4

For ease of calculation, let's say there are only 20 letter keys to chose from to be used with modifier. e.g.

- 1
`Ctrl`+`a` - 2
`Ctrl`+`b` - 3
`Ctrl`+`c` - ...
- 20
`Ctrl`+`t`

So the total is

C[4,1] = 4 4 * 20 = 80 # 80 possibilities of 4 modifiers holding 1 down, with 20 letter choices

If we can hold 2 modifiers, we have

`Alt`+`Ctrl``Alt`+`Shift``Alt`+`❖ Window``Ctrl`+`Shift``Ctrl`+`❖ Window``Shift`+`❖ Window`

or by a formula

C[4,2] = 6 # multiply by 20 letter key choices 6 x 20 = 180

The formula C[n,k] is
the possibilities of taking k things out of n.
( The formula is usually called Binomial coefficient, and is
`Binomial[n,k] := n! / (k! (n - k)! )`

)

Continue on, the total possibilities is

C[4,1] = 4 # 4 modifiers, holding 1 C[4,2] = 6 # 4 modifiers, holding 2 C[4,3] = 4 # 4 modifiers, holding 3 C[4,4] = 1 # 4 modifiers, holding 4 sum = 15 # multiply by 20 letter key choices 15 x 20 = 300

So 4 modifiers, holding 1 to 4 of them at the same time, plus a letter key give us 300 possibilities.

if we continue to count modifier by left/right, we have 8 modifiers:

C[8,1] = 8 # 8 modifiers, holding 1 C[8,2] = 28 # 8 modifiers, holding 2 C[8,3] = 56 C[8,4] = 70 sum = 162 162 * 20 = 3240 possibilities of chord with holding max of 4 out of 8, with 20 letter choices

8 modifiers, holding 1 to 4 of them at the same time, then press 1 of 20 letter key choices, give us 3240 possibilities.

We stop at holding 4 modifier at the same time.

### How Many Key Sequence Combinations Are There?

Now we consider key sequences such as
`F10` `e` `c`.

Let's call the first key (such as `F10`) the leader key.
Let's suppose there are 20 letter keys to chose from, for easy calculation.

For total of 3 key presses, counting the leader key, and suppose 20 letter key choices, we have 400 possibilities.

# 3 keys key sequence possibilities 1 * 20 * 20 = 400

If we allow total of 4 key strokes, we have 8000 possibilities.

# 4 keys key sequence possibilities 1 * 20 * 20 * 20 = 8000

That's a lot more than chord keys.

Now we consider leader key sequence of 3 keys, but 2 different leader keys, we have

# 3 keys key sequence, 2 possible leader keys 2 * 20 * 20 = 800 # 2 leader key

Here's a list of n leader keys possibilities:

# 3 keys key sequence, n leader keys 1 * 20 * 20 = 400 # 1 leader key to chose from 2 * 20 * 20 = 800 # 2 leader key to chose from 3 * 20 * 20 = 1200 4 * 20 * 20 = 1600 5 * 20 * 20 = 2000 6 * 20 * 20 = 2400 7 * 20 * 20 = 2800 8 * 20 * 20 = 3200

So, 3 key sequence, with 8 possible leader keys, give us 3200 possibilities.

## Conclusion

Comparing the ease of pressing keys and possible combinations:

- key sequences can create far more shortcuts than modifier key combinations.
- key sequences are far more easier to press than modifier key combinations.

Note: the holding modifer keys are bad due to the position of the modifiers, typically at corner of keyboard, and due to a particular order of press/release required. For example, to press
`Ctrl`+`c`, you need to it in this order:

- press
`Ctrl` - press
`c` - release
`c` - release
`Ctrl`

Must be in that order precisely.

If the modifier key are designed as true chord keyboard keys, such as on piano or steno machine, then chord is faster to press than key sequences (because you press all of the keys at once, instead of one by one), and also more combinations (because now you can press practically 8 keys at once (as in piano playing).)

See also:

- patreon me $5
- amazon egift card to xah@xahlee.org , $20 is nice.
- paypal to xah@xahlee.org , $20 is nice.
- bitcoin me 19dfoa3Q7oehm9MwCULQzBG8vqfCaeMazH

If you have a question, put $5 at patreon and message me.