- 5 persons to be seated on 5 chairs arranged in a row. Two of these persons cannot sit next to each other. How many seating arrangements are possible?
- How many four-digit integers for which the thousands digit equals the sum of the other three digits?
- How many different 5-digit numbers can be obtained by using any 5 of the digits: 1, 3, 5, 5, 5 and 5?
- You are to form a sequence of three numbers from the list 0, 1, 2, 3, 4, 5 and 6. Repetition is allowed, such as 0, 0, 6. How many possible lists can be formed such that the sum of the elements in the list is equal to 6?
- A garage door opener has a ten-digit keypad. Codes to open the door must consist of 6 digits with no adjacent digits the same. How many codes are possible?
- A number is multy if one of its digits is the product of other digits, such as 263. How many three-digit numbers are multy?
- The following sequence of letters: $$A,A,B,A,B,C,A,B,C,D,A,B,C,D,E,A,B,C,D,E,F,A …$$ is formed by writing the first letter of the alphabet, followed by writing the first two letters of the alphabet, and continuing the pattern by writing one more letter of the alphabet each time. Continuing this pattern, what letter is the 284th letter in this sequence?
- Start at S in the middle and form a path by moving to an adjacent letter to the right, left, up or down. How many paths spell the word SCHOOL ?
L
L O L
L O O O L
L O O H O O L
L O O H C H O O L
L O O H C S C H O O L
L O O H C H O O L
L O O H O O L
L O O O L
L O L
L
- A party is attended by 11 politicians and 5 lawyers. Each politician shook hands exactly once with everyone. Each lawyer shook hands exactly once with each politician, but no handshakes among lawyers. How many handshakes took place?
- A hexagon is inscribed in a circle. What is the maximum number of non-overlapping regions inside the circle divided by the edges and diagonals of the hexagon? How
Answers
- 72
- 219
- 30
- 28
- 590,490
- 52
- H
- 124
- 110
- Let $n$ is the number of side of the polygon inscribed in the circle. For $n=0$, there is one region of the circle when no chords have been created. Any time we connect two points out of $n$ points, one region is added. In this way $n\choose 2$ regions are added by the edges and diagonals. However, the line segment which connects any point with another point may intersect previously-drawn line segments creating new regions; there are $n\choose 4$ added by these intersections, as four points determine one additional region. In total, the maximum number of regions is: $$1+\dfrac{n(n-1)}{2}+\dfrac{n(n-1)(n-2)(n-3)}{4!}$$ When $n=6$, the answer is $31$.
