How Many Keyboard Shortcuts Are There

By Xah Lee. Date: . Last updated: .

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

  1. Alt
  2. Ctrl
  3. Shift
  4. ❖ 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.

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

  1. Alt+Ctrl
  2. Alt+Shift
  3. Alt+❖ Window
  4. Ctrl+Shift
  5. Ctrl+❖ Window
  6. 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

So 8 modifiers, holding 1 to 4 of them at the same time, plus 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
# 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 a leader key, 20 letter key choices, 8 possible leader keys, we have 3200 possibilities.

Conclusion

Comparing the ease of pressing keys and possible combinations, leader key beats modifiers combinations.

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