The primitive of multiplicity. How many?
Question 1: You have 5 different reagent bottles. How many ways can you arrange them on a shelf?
Question 2: A protein has 100 amino acid positions. Each position can be one of 20 amino acids. How many possible sequences exist?
Question 3: Benzene has 6 equivalent positions. You want to substitute 2 of them with chlorine. How many dichlorobenzene isomers are there?
These questions share a structure: counting configurations.
The numbers involved grow rapidly. Question 2 has more possible sequences than atoms in the observable universe.
We need systematic methods.
COLLECTION: "There are many."
Before we can do anything with objects — arrange them, select them, distribute them — we must count them.
Counting is the foundation. The tools are:
The most fundamental counting rule.
If task A can be done in m ways, and for each way of doing A, task B can be done in n ways, then A followed by B can be done in m × n ways.
Watch how choices multiply at each step.
License plates: 3 letters followed by 3 digits
DNA sequences of length k:
Each position has 4 choices (A, T, G, C). For k positions: 4k sequences.
General formula: n choices at each of k positions → nk sequences.
How many ways to arrange n distinct objects in a row?
The factorial of a non-negative integer n is:
$$n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$$
with the convention that 0! = 1.
Factorials grow faster than exponentials. 70! > 10100 (a googol).
For large n:
$$n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$$
Or more simply: $$\ln(n!) \approx n\ln(n) - n$$
| n | n! |
|---|---|
| 0 | 1 |
| 5 | 120 |
| 10 | 3,628,800 |
| 20 | 2.43 × 1018 |
| 100 | 9.33 × 10157 |
A permutation is an arrangement of objects where order matters.
From n distinct objects, the number of ways to select and arrange k of them:
$$P(n, k) = \frac{n!}{(n-k)!} = n \times (n-1) \times \cdots \times (n-k+1)$$
Example: From 10 runners, how many ways to award gold, silver, bronze?
P(10, 3) = 10 × 9 × 8 = 720
How many arrangements of the letters in MISSISSIPPI?
The formula:
$$\frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}$$
MISSISSIPPI: $\frac{11!}{1! \times 4! \times 4! \times 2!} = \frac{39,916,800}{1,152} = 34,650$
A combination is a selection of objects where order does not matter.
From n distinct objects, the number of ways to choose k of them:
$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$
Read as "n choose k"
Key insight: Each unordered selection of k objects can be arranged in k! ways. So:
$$\binom{n}{k} = \frac{P(n,k)}{k!}$$
Symmetry: $\binom{n}{k} = \binom{n}{n-k}$
Choosing k objects to include is equivalent to choosing n−k objects to exclude.
Pascal's Identity: $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$
Each entry is the sum of two entries above it.
Sum of row n: $\displaystyle\sum_{k=0}^{n} \binom{n}{k} = 2^n$
$$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$
Example: $(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$
Coefficients: 1, 4, 6, 4, 1 — row 4 of Pascal's triangle.
| Situation | Formula | When to use |
|---|---|---|
| n choices at each of k positions | nk | Independent choices with replacement |
| Arrange n distinct objects | n! | All objects, order matters |
| Arrange k of n distinct objects | n!/(n−k)! | Subset, order matters |
| Arrange n with repetitions | n!/(n₁!n₂!···) | Identical objects present |
| Choose k of n objects | n!/[k!(n−k)!] | Subset, order doesn't matter |
How many structural isomers of C₅H₁₂ (pentane)?
Not a simple formula — requires enumerating distinct carbon skeletons:
Answer: 3 isomers.
For larger alkanes, the count grows rapidly:
Benzene has 6 equivalent positions. Choose 2 for Cl.
Naive count: $\binom{6}{2} = 15$
But benzene has symmetry. Positions related by rotation/reflection give the same isomer.
Actual distinct dichlorobenzene isomers:
Answer: 3 isomers. Symmetry reduces the count from 15 to 3.
How many ways to put electrons in orbitals?
For p² (2 electrons in p subshell): C(6, 2) = 15 microstates
These 15 microstates span three term symbols: ³P (9), ¹D (5), ¹S (1).
How many possible sequences for a 100-residue protein?
20 amino acid choices at each of 100 positions:
$20^{100} = 10^{130}$
For comparison:
The sequence space of even modest proteins is astronomically vast. This is why evolution explores only a tiny fraction of possible sequences.
The Boltzmann entropy:
$$S = k_B \ln W$$
where W is the number of microstates.
If n₁ particles have energy E₁, n₂ have E₂, etc., the number of arrangements is:
$$W = \frac{N!}{n_1! n_2! n_3! \cdots}$$
This multinomial coefficient counts how many ways to partition N distinguishable particles into groups.
(a) How many 5-letter "words" can be formed from the 26-letter alphabet (letters can repeat)?
(b) How many if no letter can repeat?
(c) How many 5-letter words have exactly one vowel (A, E, I, O, U)?
(a) Compute 8!/5! without computing each factorial separately.
(b) Simplify: n!/(n−2)!
(c) Use Stirling's approximation to estimate ln(50!).
(a) How many ways can 8 people stand in a line?
(b) How many ways can 8 people sit at a round table? (Rotations are considered the same.)
(c) How many distinct arrangements of the letters in CHEMISTRY?
(a) From a standard 52-card deck, how many 5-card hands are possible?
(b) How many of these hands contain exactly 2 aces?
(c) How many contain at least one ace?
(a) How many tripeptides can be formed from the 20 standard amino acids?
(b) Methane (CH₄) has 4 equivalent H atoms. How many distinct mono-chlorinated products exist when one H is replaced by Cl?
(c) For the complex [Co(NH₃)₄Cl₂]⁺, how many geometric isomers exist? (Hint: octahedral geometry)