Permutations & Combinations Question - None

Question

If $\lambda$ be the number of 3-digit numbers of the form $xyz$ with $x < y$, $z < y$ and $x \neq 0$, find the value of $\frac{\lambda}{30}$.

Accepted Answer

Solution:

Since $x \geq 1$ and $x < y$, we have $y \geq 2$.

For each value of $y$ (where $2 \leq y \leq 9$):

- $x$ takes values from 1 to $y-1$ (i.e., $y-1$ choices)

- $z$ takes values from 0 to $y-1$ (i.e., $y$ choices)

Thus, for each value of $y$, there are $(y-1) \cdot y$ valid pairs $(x,z)$.

Total: $\lambda = \sum_{y=2}^{9} y(y-1) = 2(1) + 3(2) + 4(3) + \cdots + 9(8) = 240$

Therefore, $\frac{\lambda}{30} = \frac{240}{30} = 8$

Question: