?

You and two friends visit a small screening of the movie "Logan".

The movie theatre consists of 10 rows each with 10 seats and is 3/4 full when you and your friends arrive.

Assuming that the 75 people are randomly seated, what is the probability that you and your friends can find 3 adjacent empty seats in the same row so that you may all sit together?

I am happy with an estimate but would prefer an exact fraction if possible.

Good luck!

Hint: This number has prime factorisation 2^5 * 3^8 * 5^6 * 7^3 * 11^5

5 people, each of a different height from the others, queue to buy tickets at the movies. What is the probability that just 1 person in the queue is taller than the individual standing directly in front of them?

(a) How many permutations of the letters in MISSISSIPPI are there with no adjacent identical letters?

(b) How many permutations of the letters in WALLAWALLA are there that don't contain the subwords AAA, LLL or WW?

I'm thinking 4 points: those of a rectangle. The hint in the question says to use "Fubini's Principle or otherwise".

a) 1

b) 0

c) 1/2

Please justify your answer.

How many distinct arrangements are there of the letters in NASHVILLETENNESSEE with the first N preceeding the first S and the first E preceeding the T?

To re-iterate: the grid of dots is 10 dots by 10. They are equally spaced and the squares formed can be any size and any orientation.

A balance scale sits on a teacher's table, currently tipped to the right. There is a set of weights on the scales and on each weight is the name of at least one pupil.

On entering the classroom, each pupil moves all the weights carrying his or her name to the opposite side of the scale.

Prove that there is *some* set of pupils that you, the teacher, can let in which will tip the scales to the left.

For n∈N, how many ways can n +1's and n -1's be arranged such that their sum from the left is ≥0 **at all points**?

eg. For n=3, +1+1-1-1+1-1 is one such arrangement because:

+1≥0

+1+1≥0

+1+1-1≥0

+1+1-1-1≥0

+1+1-1-1+1≥0

+1+1-1-1+1-1≥0

How do you prove that *every* subset of {1,2,3,...,98,99,100} which contains exactly 10 elements must itself contain at least 2 disjoint non-empty subsets with the same sum.

*************

eg. The set {1,5,24,25,45,67,70,78,84,90} contains the two subsets {1,24} and {25} which are disjoint (no common elements) and have the same sum of 25.

How many arrangements of the digits 0-9 are there so that each digit (except the first) differs from some digit on it's left by 1?

I have 4 coloured pens in my pencil case: 2 blue, 1 red and 1 green. I reach in an pick 2 pens out at random, if one of the pens is blue what is the probability that the second is also blue?

Mary has 3 boxes of cupcakes she has just bought. Showing the boxes to her husband Joe she tells him: "In one of the boxes there are 2 chocolate cupcakes, in another there is 1 chocolate and 1 vanilla and in the third box there are 2 vanilla cupcakes."

She then picks out a box at random, removes from it a chocolate cupcake and says to Joe: "If you can tell me the probability that the other cupcake in this box is also chocolate you can have any cupcake you like."

What should he answer?

A contestant on a game show is given a choice of 3 boxes A, B and C. Two of the boxes are empty but one of them contains £1000.

The contestant chooses a box at random, but before she can open it the game show host stops her and says: "I will now remove one of the empty boxes from the two that you didn't choose and you may decide whether to swap your current choice for the remaining box."

What should the contestant do: Stick with her selection or swap it for the other box? Or doesn't it matter either way?

A regular analogue clock has a 'minute' hand and an 'hour' hand which rotate at constant angular speeds - 1 rev per hour and 1/12 rev per hour respectively.

How many times between 3:00 pm and 12:00 am can the the two hands be swapped so that the clock face displays another time that is also inside this 9 hour time period?

What are those times?

****************************

Hint:

Clearly 3:30 is *not* such a time as there is no such time where the 'minute' hand points *directly* at the 3 whilst the 'hour' hand points *directly* at the 6.

-A wooden cube of uniform density has sides of length 12 cm and a mass of 1.296 kg.

-A circular hole, 4 cm in diameter, is drilled into the middle of each face of the cube.

-Each hole is drilled perpendicular to the face and tunnels all the way through to the opposite side of the block.

-The end result is a block of wood with 3 perpendicular intersecting tunnels through it's centre.....oh, and a pile of sawdust on the floor.

The puzzle is this:

What is the mass of the drilled block of wood?

So, this question was given to me years ago when I was still at school. I am still not sure how to solve it so here goes...

You have a block of cheese in a tight fitting perspex container.You paint the outside of the cheese black and then divide it into 27 equal cubes (3x3x3) then randomize the cubes in a bag. You drop them back into the container directly from the bag. What is the probability that they will land so that all 6 faces of the block of cheese display entirely black faces?