The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word 'SYLLABUS' such that two letters are distinct and two letters are a like, is _________.

The natural number m, for which the coefficient of x in the binomial expansion of ${\left({\mathrm{x}}^{\mathrm{m}} + \frac{1}{{\mathrm{x}}^2}\right)}^{22} \mathrm{is} 1540, \mathrm{is}$

23.

If the line, 2x – y + 3 = 0 is at a distance $\frac{1}{\sqrt{5}} \mathrm{and} \frac{2}{\sqrt{5}} \mathrm{from} \mathrm{the} \mathrm{lines} 4\mathrm{x} - 2\mathrm{y} + \mathrm{\alpha } = 0 \mathrm{and} 6\mathrm{x} - 3\mathrm{y} + \mathrm{\beta } = 0,$

hen the sum of all possible values of $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ is_

$\mathrm{Let} \mathrm{f}\left(\mathrm{x}\right) = \mathrm{x}, \left[\frac{\mathrm{x}}{2}\right] \mathrm{for} –10 < \mathrm{x} < 10, \mathrm{where} \left[\mathrm{t}\right] \mathrm{denotes} \mathrm{the} \mathrm{greatest} \mathrm{integer} \mathrm{function}. \mathrm{Then} \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{points} \mathrm{of} \mathrm{discontinuity} \mathrm{of} \mathrm{f} \mathrm{is} \mathrm{equal} \mathrm{to} \mathrm{\_\_\_\_}$

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is _______