found in Figure 7a is shown in Figure 7b, in which asterisks are eliminated. You will note that the number of different combinations is 6.
The formula for obtaining the number of different combinations of n objects of which the number p are of one kind, q of another, r of another, and so on, involves a division of factorials, thus:
| n! |
| p! x q! x r! . . . . . |
In the case we have just cited—that is, the four-amino-acid protein with two amino acids of one type and two of another—the formula is:
| 4! | or | 4 x 3 x 2 x 1 | or 6 |
| 2! x 2! | 2 x 1 x 2 x 1 |
Of course, the factorials involved in calculating the number of amino acid combinations in hemoglobin are larger. We must start with factorial 539—the total number of amino acids in hemoglobin—and divide that by the product of factorial 75, factorial 54, factorial 50 and so on—the number of each amino acid present.
| 1! | equals | 1 | equals | 1 |
| 2! | equals | 2 x 1 | equals | 2 |
| 3! | equals | 3 x 2 x 1 | equals | 6 |
| 4! | equals | 4 x 3 x 2 x 1 | equals | 24 |
| 5! | equals | 5 x 4 x 3 x 2 x l | equals | 120 |
| 6! | equals | 6 x 5 x 4 x 3 x 2 x l | equals | 720 |
| 7! | equals | 7 x 6 x 5 x 4 x 3 x 2 x l | equals | 5040 |
| 8! | equals | 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 | equals | 40320 |
|
Figure 8. The factorials of the first few integers. | ||||
The factorials of the lower integers are easy enough to calculate (see Figure 8). Unfortunately they build up rather rapidly. Would you make a quick guess at the value of factorial 20? You're probably wrong. The answer is approximately twenty-four hundred quadrillion, which, written in figures, is 2,400,000,000,000,000,000. And factorial values continue mounting at an ever-increasing rate.
In handling large numbers of this sort, recourse is had to exponentials of the form 10n. 10n is a short way of representing a numeral consisting of 1 followed by n zeros. 1,000 would be 103 and 1,000,000,000,000 would be 1012 and so on. A number like 2,500 which is in between 1,000 (that is 103) and 10,000 (that is 104) could be expressed as 10 to a fractional exponent somewhere in between 3 and 4. More often, it is written simply as 2.5 x 103 (that is, 2.5 x 1,000—which, obviously, works out to 2,500).
Written exponentially, then, factorial 20 is about 2.4 x 1018.
For the purposes of this article, there are several things that must be kept in mind with regard to exponential numbers:
1) In multiplying two exponential numbers, the exponents are added.
Hemoglobin and the Universe
135
