Calculating the Chances of a Random Arrangement

The number of possible arrangements of a given number of objects increases dramatically with the addition of more objects. In fact, the numbers can be so staggering, we show on this page why the mathematical formula for determining the number of arrangements is accurate.

The formula to determine how many possible arrangements there are for a given number of objects is to multiply progressive numbers from 1 to the number of objects.

For instance, the mathematical formula to determine how many possible arrangements there are for two cards is 1x2=2. There are two possible arrangements of two cards. They are:

A K

K A

The formula for the number of possible arrangements of 3 cards is 1x2x3=6 because with the addition of the third card there are three times as many combinations as before. So, there are 6 possible arrangements for three cards. They are:

A K Q

A Q K

K A Q

K Q A

Q A K

Q K A

The formula for the number of possible arrangements of 4 cards is 1x2x3x4=24. So, there are 24 possible arrangements of 4 cards because with the addition of the 4th card, there are 4 times as many different arrangements as before. They are:

A K Q J

A K J Q

A Q K J

A Q J K

A J K Q

A J Q K

K A Q J

K A J Q

K Q A J

K Q J A

K J A Q

K J Q A

Q A K J

Q A J K

Q K A J

Q K J A

Q J A K

Q J K A

J A K Q

J A Q K

J K A Q

J K Q A

J Q A K

J Q K A

The formula for the number of possible arrangements of 5 cards is 1x2x3x4x5=120 because with the addition of the 5th card, there are 5 times as many arrangements than there were before. There are 120 possible arrangements of only 5 cards. They are:

A K Q J 10

A K J Q 10

A Q K J 10

A Q J K 10

A J K Q 10

A J Q K 10

A K Q 10 J

A K J 10 Q

A Q K 10 J

A Q J 10 K

A J K 10 Q

A J Q 10 K

A K 10 Q J

A K 10 J Q

A Q 10 K J

A Q 10 J K

A J 10 K Q

A J 10 Q K

A 10 K Q J

A 10 K J Q

A 10 Q K J

A 10 Q J K

A 10 J K Q

A 10 J Q K

K A Q J 10

K A J Q 10

K Q A J 10

K Q J A 10

K J A Q 10

K J Q A 10

K A Q 10 J

K A J 10 Q

K Q A 10 J

K Q J 10 A

K J A 10 Q

K J Q 10 A

K A 10 Q J

K A 10 J Q

K Q 10 A J

K Q 10 J A

K J 10 A Q

K J 10 Q A

K 10 A Q J

K 10 A J Q

K 10 Q A J

K 10 Q J A

K 10 J A Q

K 10 J Q A

Q A K J 10

Q A J K 10

Q K A J 10

Q K J A 10

Q J A K 10

Q J K A 10

Q A K 10 J

Q A J 10 K

Q K A 10 J

Q K J 10 A

Q J A 10 K

Q J K 10 A

Q A 10 K J

Q A 10 J K

Q K 10 A J

Q K 10 J A

Q J 10 A K

Q J 10 K A

Q 10 A K J

Q 10 A J K

Q 10 K A J

Q 10 K J A

Q 10 J A K

Q 10 J K A

J A K Q 10

J A Q K 10

J K A Q 10

J K Q A 10

J Q A K 10

J Q K A 10

J A K 10 Q

J A Q 10 K

J K A 10 Q

J K Q 10 A

J Q A 10 K

J Q K 10 A

J A 10 K Q

J A 10 Q K

J K 10 A Q

J K 10 Q A

J Q 10 A K

J Q 10 K A

J 10 A K Q

J 10 A Q K

J 10 K A Q

J 10 K Q A

J 10 Q A K

J 10 Q K A

10 A K Q J

10 A Q K J

10 K A Q J

10 K Q A J

10 Q A K J

10 Q K A J

10 A K J Q

10 A Q J K

10 K A J Q

10 K Q J A

10 Q A J K

10 Q K J A

10 A J K Q

10 A J Q K

10 K J A Q

10 K J Q A

10 Q J A K

10 Q J K A

10 J A K Q

10 J A Q K

10 J K A Q

10 J K Q A

10 J Q A K

10 J Q K A

The formula for the number of possible arrangements of 6 cards is 1x2x3x4x5x6=720 because with the addition of the 6th card there are 6 times as many arrangements as there were with only 5 cards. So, just 6 cards can be arranged in 720 possible ways.

The number of possible arrangements of 20 cards is 1x2x3x4x5x6...x20 = 2,432,902,008,176,640,000.

So the chances of accidentally dealing 20 cards in one particular order is 1 in 2,432,902,008,176,640,000. Given that, what are the chances of just 36 different enzymes being accidentally arranged in the correct order by chance mutations in order to create one necessary protein or other necessary molecule? That number, whatever it is, is the reason why it did not happen by accident.