Careful Counting
In how many ways can you line up 5 boys and 5 girls if two kids of the same gender cannot sit next to each other? You may leave your answer as an expression.
2*5!*5! = 28,800
How many 10-digit numbers have the sum of their digits equal to 3?
55
What is the remainder of 12 + 22 + 32 + . . . + 262 (mod 5)
1
What are the last two digits of 512 - 262?
25
Betsy the Bug has n pairs of feet, where n is a positive integer. Betsy also has an unlimited supply of green, blue, and red shoes. For a given pair of feet, Betsy can either be barefoot on both feet, or wear shoes on both feet, in which case she can mix-and-match shoe colors. How many ways can Betsy choose to dress her feet? Your answer will be an expression involving n.
10n
In a hard fought hockey match, the New York Rangers defeated the Pittsburgh Penguins 5-2. How many different sequences of goals were possible if it is known that the Rangers never trailed?
14
What are the last three digits of 9989?
488
Here is a list of months, and a code for each month:
January: 7110
February: 826
March: 5313
April: 541
May: 3513
June: 4610
July: 4710
What is the code for the month of August?
681
The MMA Teachers Club is a group of 8 women and 6 men. Two of the men, Arnav and Bobby, are feuding and refuse to speak to each other. How many ways can we select 3 of the women and 2 of the men to serve on a committee if Arnav and Bobby refuse to serve on the committee together?
784
Ten points are marked on a straight line and 11 points are marked on another line, parallel to the first one. How many triangles are there with vertices (corners) at these points?
1045
The three smallest 5-digit prime numbers are 10007, 10009, and 10037.
If they are multiplied together and divided by 9, what is the remainder?
7
Evaluate:
1002 - 992 + 982 - 972 + . . . + 42 - 32 + 22 - 12.
5050
How many divisors of 9,000 are not perfect squares?
40
What is the smallest positive integer with exactly 32 divisors?
840
What is the number 456,564,465,645 congruent to (mod 6)?
3
All positive integers which only use the digits 1, 3, 5, and 7 are written in an increasing order (1, 3, 5, 7, 11, 13, 15, 17, . . . ). What is the 90th number written?
1133