My wallet $7 lighter, I walked out of the store that sells used books with a book of math puzzles. I'll give one of the easiest ones from the book, plus a medium-difficult one. Maybe later, a difficult one.

I.
A wooden cube has edges of length 3 meters. Square holes of side one meter, centered in each face, are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside is how many square meters?

II.
A box contains 2 pennies, 4 nickels and 6 dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chose. What is the probability that the value of the coins drawn is at least 50 cents?

[For those outside the USA, penny = 1 cent, nickel = 5 cents, dime = 10 cents.]

Originally posted by Paul Dirac II My wallet $7 lighter, I walked out of the store that sells used books with a book of math puzzles. I'll give one of the easiest ones from the book, plus a medium-difficult one. Maybe later, a difficult one.

I.
A wooden cube has edges of length 3 meters. Square holes of side one meter, centered in each face, are cut through to the opposite face. The ...[text shortened]... ast 50 cents?

[For those outside the USA, penny = 1 cent, nickel = 5 cents, dime = 10 cents.]

Originally posted by Paul Dirac II The book gives the answer as being a substantially higher probability than that. How do you justify the 4 in the numerator of your first line?

I suppose Iv'e overlooked something. I assumed there were only 4 favorable cases where the sum of the 6 coins is >= 50 cents

It might help to briefly consider an alternative problem where the box contains two pennies, four nickels, and a million dimes. Intuitively, the probability of totaling at least 50 cents when pulling six coins out randomly from such a box is almost 100%.

Another thing that might help in the case of the box of 2 + 4 + 6 coins is to imagine some sneaky person has penciled labels on the coins. For example, one of the dimes might have 'A' written on it, another dime might have 'B' written on it, and so on.

I get:
12 coins
number of ways of drawing 6 coins = 12*11*10*9*8*7

there are 4 possible solutions:
6d
1n 5d
1p 5d
2n 4d

first, consider how many different combinations with 'labelled coins' there are

With 6d we have used all our dimes, so only one combination (permutations come later)
with 1n 5d,we can have 1 out of 4 nickels and miss out any one dime, so 4*6 combinations
similarly for 1p 5d, 2*6 combinations

2n 4d is more complex
2n can be selected in 4*3/2 = 6 ways
2d can be selected in 6*5/2 = 15 ways
giving 6*15 combinations

so the total combinations giving a solution is:
1 + 4*6 + 2*6 + 15*6 = 1 + 21*6 = 127

each combination has 6*5*4*3*2 permutations
giving a total 127*6*5*4*3*2 solution permutations

Originally posted by iamatiger Anyone want to work out average total with 6 random coins?

Intuitively, it should be 41 cents. You're taking half the coins, so on average, you should have half your pennies, half your nickles, and half your dimes.