Discrete Mathematics

# Distribution into Bins: Level 5 Challenges

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

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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