Basic Counting Principles: Additional Practice (Set 2)

Fill the three chairs in sequence, with the choices shrinking as each person sits.

$5 \times 4 \times 3 = 60$

Award the four places in sequence, with the choices shrinking each step.

$6 \times 5 \times 4 \times 3 = 360$

Count each type with the multiplication principle: a letter then a digit gives $26 \times 10 = 260$, and a digit then a digit gives $10 \times 10 = 100$. The password is one type or the other, so add the two counts.

$260 + 100 = 360$

Fill the six positions one at a time, with the choices shrinking each step.

$6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$

Fill the five positions one at a time, with the choices shrinking each step.

$5 \times 4 \times 3 \times 2 \times 1 = 120$

Four stages in sequence with $2$, $6$, $2$, and $6$ outcomes, so multiply all four.

$2 \times 6 \times 2 \times 6 = 144$

Three independent choices in sequence, so multiply the three counts.

$3 \times 6 \times 5 = 90$

Fill the three roles in sequence, with the choices shrinking each step.

$10 \times 9 \times 8 = 720$

Each switch has $2$ states, repeated for all ten switches, so multiply ten twos.

$2^{10} = 1024$

A dress alone gives $6$ outfits. A top and a skirt is a sequence of two choices, giving $4 \times 5 = 20$. The outfit is a dress or a top-and-skirt, so add the two counts.

$6 + 20 = 26$

Choose the digits in sequence with no repeats, so the choices shrink from $10$.

$10 \times 9 \times 8 = 720$

A leading $0$ is allowed, so all $10$ digits are available for the first position.

The first position has $8$ choices and each of the next six has $10$, so multiply.

$8 \times 10^6 = 8{,}000{,}000$