Discrete Mathematics
# Distribution into Bins

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

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

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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