The number of possible outcomes when a coin is tossed 6 times is (A) 36 (B) 64 (C) 12 (D) 32

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Concept understanding — Counting Principle

The Counting Principle: From Intuition to Precision

Imagine you're ordering a pizza. You have two choices for crust — thin or thick — and three choices for topping — cheese, pepperoni, or mushroom. How many different pizzas can you make?

You could list them all: thin-cheese, thin-pepperoni, thin-mushroom, thick-cheese, thick-pepperoni, thick-mushroom. That's 6 pizzas.

Notice something: 2 crusts × 3 toppings = 6 total combinations. That's the Counting Principle in action.

The Intuition

The Counting Principle answers one simple question: If I make a sequence of choices, how many possible outcomes are there?

Think of it as building a path. At each step, you have a certain number of options. The total number of complete paths is just the product of the number of options at each step.

Why multiplication? Because for each choice at step 1, you can pair it with every choice at step 2, and so on. It's like a tree that branches out — the number of leaves at the end is the product of the number of branches at each level.

Note

The Counting Principle is also called the Fundamental Principle of Counting or the Multiplication Principle. It's the foundation of all combinatorics.

The Precise Statement

\text{If an event can occur in } m \text{ ways, and for each of these, a second event can occur in } n \text{ ways, then the two events together can occur in } m \times n \text{ ways.}

More generally: If you have $k$ steps, and step $i$ has $n_i$ possible choices, then the total number of outcomes is:

n_1 \times n_2 \times n_3 \times \cdots \times n_k

Key Conditions

The principle works only when choices at different steps are independent — meaning the number of options at one step does not depend on what you chose earlier.

Watch out

If choices are dependent (e.g., picking two people from a group without replacement), you cannot simply multiply the raw numbers. You must adjust for the dependency. That's where permutations and combinations come in later.

Examples to Lock It In

Example 1: Outfits

You have 4 shirts, 3 pants, and 2 pairs of shoes. How many outfits?

4 \times 3 \times 2 = 24

Example 2: License Plates

A plate has 3 letters followed by 3 digits. Letters can repeat, digits can repeat.

26 \times 26 \times 26 \times 10 \times 10 \times 10 = 17,576,000

Example 3: Multiple-Choice Test

A test has 5 questions, each with 4 options. How many answer patterns?

4 \times 4 \times 4 \times 4 \times 4 = 4^5 = 1024

A Common Mistake …

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Each coin toss has $2$ independent outcomes, so $6$ tosses yield $2^6 = 64$ total possible sequences.

Why the multiplication principle applies

When we toss a coin once, we get either heads or tails—exactly $2$ outcomes. The question asks: if we repeat this experiment $6$ times, how many different sequences can we observe?

The key insight is that each toss is independent. The result of the first toss doesn't constrain the second, the second doesn't constrain the third, and so on. When events are independent and we want to count all possible combined outcomes, we multiply the number of choices at each stage.

Think of it as filling six slots:

\underbrace{\_\_}_{1st} \quad \underbrace{\_\_}_{2nd} \quad \underbrace{\_\_}_{3rd} \quad \underbrace{\_\_}_{4th} \quad \underbrace{\_\_}_{5th} \quad \underbrace{\_\_}_{6th}

For each slot, we have $2$ choices (H or T). The total number of ways to fill all six slots is the product of the choices at each position.

Counting the outcomes

First toss: $2$ possibilities (H or T).
Second toss: Again $2$ possibilities, regardless of what happened in the first toss. So far we have $2 \times 2 = 4$ possible sequences: HH, HT, TH, TT. …

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Q.The number of possible outcomes when a coin is tossed 666 times is (A) 363636 (B) 646464 (C) 121212 (D) 323232

Why the multiplication principle applies

Counting the outcomes

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Q.The number of possible outcomes when a coin is tossed $6$ times is
(A) $36$
(B) $64$
(C) $12$
(D) $32$

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