Combinatorics, Product Rule / Rule of Product

The Rule of Product is a fundamental counting principle: if there are `n` ways to do one task and `m` ways to do another task, then there are `n \times m` ways to perform both tasks. Questions involve applying this rule to count sequences of choices or outcomes.

• Question

  Danny flips a coin `3` times and records the results in a row. How many different possibilities are there for this row?

  Topics:

  Combinatorics -> Product Rule / Rule of Product

• Question

  In a soccer team, a captain and a vice-captain are chosen. In how many different ways can this selection be made?

  Note: A soccer team always has `11` players.

  Topics:

  Combinatorics -> Product Rule / Rule of Product

• Question

  In a foreign country, every phone number has `7` digits, and the first digit is always different from `0`. What is the maximum number of phone numbers that can exist in that country?

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  Topics:

  Arithmetic Combinatorics -> Product Rule / Rule of Product
  Sources:
  • problems.ru web site

• Question

  We call a number nice if it consists only of odd digits. How many four-digit nice numbers exist?

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  Topics:

  Arithmetic Combinatorics -> Product Rule / Rule of Product
  Sources:
  • problems.ru web site

• Question

  Alice loves to drink tea. For this, she always takes one cup and one saucer for the cup. Alice has `3` different cups and `2` different saucers. In how many different ways can she assemble her tea set?

  Topics:

  Combinatorics -> Product Rule / Rule of Product

• Question

  From city A' to city B', there are `4` roads, and from city B' to city C', there are `3` roads. In how many different ways can one get from A' to C', if we are only allowed to travel on the roads, and in addition, we do not want to travel backwards? 

  Topics:

  Combinatorics -> Product Rule / Rule of Product

• Question

  In a class there are `15` girls and `16` boys. We want to choose a committee consisting of one girl and one boy. In how many different ways can the committee be chosen?

  Topics:

  Combinatorics -> Product Rule / Rule of Product

• Question

  In a company, there are `8` mathematicians, `15` economists, and `25` programmers. We need to choose a committee of three people, including one person from each profession. In how many different ways can the committee be chosen?

  Topics:

  Combinatorics -> Product Rule / Rule of Product

• Question

  Given a square that is divided into `4` smaller squares. In how many different ways can the smaller squares be colored green and orange?

  Topics:

  Combinatorics -> Product Rule / Rule of Product

• Question

  Are there more seven-digit numbers that have no zeros in their decimal representation, or those that have no two adjacent identical digits in their decimal representation?

  Topics:

  Combinatorics -> Product Rule / Rule of Product