12 multiple-choice questions, progressively harder.
A license code is 222 letters followed by 222 digits, and repeats are allowed. How many codes are possible? (There are 262626 letters and 101010 digits.)
Solution
Correct answer: D
Four positions in sequence, with 262626 choices for each letter and 101010 for each digit, so multiply all four.
26×26×10×10=6760026 \times 26 \times 10 \times 10 = 6760026×26×10×10=67600
How many 333-digit numbers have all three digits different? (A 333-digit number cannot start with 000.)
Correct answer: C
The first digit is 111 through 999, so 999 choices. The second digit may be any digit except the first (999 left, since 000 is now allowed), and the third any digit except the first two (888 left).
9×9×8=6489 \times 9 \times 8 = 6489×9×8=648
A coin is flipped 777 times. How many different sequences of heads and tails are possible?
Correct answer: A
Each flip has 222 outcomes, repeated for all seven flips, so multiply seven twos.
27=1282^7 = 12827=128
A 444-digit code is formed where the first digit cannot be 000 but the other three digits can be anything 000 through 999 (repeats allowed). How many codes are possible?
The first position has 999 choices (111 through 999) and each of the other three has 101010, so multiply.
9×10×10×10=90009 \times 10 \times 10 \times 10 = 90009×10×10×10=9000
How many 333-digit even numbers are there? (A 333-digit number cannot start with 000, and an even number ends in 0,2,4,6,0, 2, 4, 6,0,2,4,6, or 888.)
Correct answer: B
Choose the digits in sequence. The first digit is 111 through 999 (999 choices), the middle is any digit (101010), and the last must be even (555 choices).
9×10×5=4509 \times 10 \times 5 = 4509×10×5=450
A club of 888 members chooses a president and a vice-president (two different people). In how many ways can this be done?
Choose the president first, then the vice-president from those who remain, with the choices shrinking.
8×7=568 \times 7 = 568×7=56
A flag has 444 horizontal stripes, each a different color chosen in order from 666 available colors. How many different flags are possible?
Color the stripes one at a time, with no color repeated, so the choices shrink.
6×5×4×3=3606 \times 5 \times 4 \times 3 = 3606×5×4×3=360
From 666 different trophies, you place 222 of them in order on a shelf (a left spot and a right spot). In how many ways can this be done?
Fill the left spot, then the right spot from those that remain, so the choices shrink.
6×5=306 \times 5 = 306×5=30
A lunch is either a sandwich with a drink (444 sandwiches, 333 drinks) or a salad with a drink (555 salads, 333 drinks). How many different lunches are possible?
Count each kind of lunch by multiplying: sandwich lunches give 4×3=124 \times 3 = 124×3=12 and salad lunches give 5×3=155 \times 3 = 155×3=15. A lunch is one kind or the other, so add the two counts.
12+15=2712 + 15 = 2712+15=27
Which gives more possible outcomes: flipping a coin 666 times, or rolling a die 222 times?
Flipping a coin 666 times gives 262^626 outcomes, while rolling a die twice gives 626^262 outcomes.
26=64and62=362^6 = 64 \quad \text{and} \quad 6^2 = 3626=64and62=36
Since 64>3664 > 3664>36, the coin flips give more outcomes.
In how many different orders can the 555 distinct letters of the word MONEY be arranged?
All five letters are different, so fill the five positions one at a time with shrinking choices.
5×4×3×2×1=1205 \times 4 \times 3 \times 2 \times 1 = 1205×4×3×2×1=120
A phone PIN has 666 digits, each independently 000 through 999 (repeats allowed). How many PINs are possible?
Each of the six positions has 101010 choices, repeated for all six, so multiply six tens.
106=1,000,00010^6 = 1{,}000{,}000106=1,000,000
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