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  4. The number of three-digit numbers of the form x y z such that x<y and z ≤ y is

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Q. The number of three-digit numbers of the form $x y z$ such that $x
Permutations and Combinations

Solution:

If zero is included it will be at $z \Rightarrow { }^{9} C _{2}$
If zero is excluded $\begin{cases} x , y , z \text { all diff. } & \Rightarrow { }^{9} C _{3} \times 2 ! \\ x = z < y & \Rightarrow { }^{9} C _{2} \\ x < y = z & \Rightarrow { }^{9} C _{2}\end{cases}$
The total number of ways is $276$ .
Alternate method :
$y$ can be from $2$ to $9$ ; so total number of ways is
$\displaystyle\sum_{ r -2}^{9}\left( r ^{2}-1\right)=276$