1.

If lamda be the number of 3-digit numbers are o the form xyz with x lt y, z lt y and x ne0, the value of (lamda)/(30) is

Answer»

Solution :Since, `x ge1, ` then `x ge 2""[becausex LTY]`
If y=n, then x takes values form 1 to n-1 and zcan TAKE the values from 0 to n-1 (i.e., n values)
thus, for each values of `y(2 le y le9), x and z` take `n(n-1)`
vlaues.
Hence, the 3-digit NUMBERS are of the form xyz
`=underset(n=2)overset(9)(sum)n(n-1)=underset(n=1)overset(9)(sum)n(n-1)""[because" at "n=1,n(n-1)=0]`
`=underset(n=1)overset(9)(sum)n^(2)-underset(n=1)overset(9)(sum)n`
`=(9(9+1)(18+1))/(6)-(9(9+1))/(2)`
`=285-45`
`=240=LAMDA""[given]`
`THEREFORE(lamda)/(30)=8`


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