Probability

Distribution into Bins

Distribution into Bins: Level 5 Challenges

         

A×B×C×D×E×F=7×107 \large A \times B \times C \times D \times E \times F = 7 \times 10^7

Determine the number of ordered solutions for integers A,B,C,D,E,FA,B,C,D,E,F such that they satisfy the equation above.

Inspiration.

How many 6-digit numbers can be formed using exactly 4 different digits?

We want to create a divisible sequence of length ll starting from a number NN. In a Divisible Sequence, every term (except the starting number) is a divisor of the previous term. Examples of divisible sequences of length 3 starting with 10 are:

10,10,1010,10,10

10,10,510,10,5

10,10,110,10,1

10,5,510,5,5 ... etc.

Find the number of divisible sequences of length 5 starting from the number 360.

Seven identical eggs are being put into a 15×115 \times 1 egg box. However no more than two eggs can be put in a row. For example, one possible arrangement is E-E----EE-E--EE.

How many possible arrangements are there for the eggs?

Note: The egg box always stays in the same orientation so reflections count.

Let p(n) p(n) be the number of partitions of n n. Let q(n)q(n) be the number of partitions of 2n 2n into exactly nn parts. For example, q(3)=3q(3) = 3 because 6=4+1+1=3+2+1=2+2+2. 6 = 4+1+1 = 3+2+1 = 2+2+2. Compute p(12)q(12). p(12)-q(12).

Definition: A partition of an integer is an expression of the integer as a sum of one or more positive integers, called parts. Two expressions consisting of the same parts written in a different order are considered the same partition ("order does not matter").

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