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Distribution into Bins

How can five candies be distributed among 3 friends? Well, the answer depends on if you can tell the candies apart!

Level 5

         

\[ \large A \times B \times C \times D \times E \times F = 7 \times 10^7 \]

Determine the number of ordered solutions for integers \(A,B,C,D,E,F\) 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 \(l\) starting from a number \(N\). 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:

\(10,10,10\)

\(10,10,5\)

\(10,10,1\)

\(10,5,5\) ... etc.

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

Seven identical eggs are being put into a \(15 \times 1 \) 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 \( p(n) \) be the number of partitions of \( n\). Let \(q(n) \) be the number of partitions of \( 2n \) into exactly \(n \) parts. For example, \(q(3) = 3 \) because \[ 6 = 4+1+1 = 3+2+1 = 2+2+2. \] Compute \( p(12)-q(12). \)

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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