Question

If combinations of letters be formed by taking only 5 letters at a time out of the letters of the word "METAPHYSICS". In how many of them will letter $T$ occur?

Answer 1

There are 11 letters T,S,S and 8 all different since T is a must we have to select only 4 out of remaining S,S and 8 all different. Mind we have to find only combination are not words (arrangements not be considered)
(a) $2$ alike $(S,S),2$ different $={{}^{2}C}_1.{{}^{8}C}_2=28$
(b) $1S,3$ different $={{}^{8}C}_3=56$
(c)No $S,4$ different ${{}^{8}C}_4=70$
For (b), (c)
Now (b) and (c) would mean that we have to select 4 out of 9 different letters ${{}^{9}C}_4=\frac{\mathrm{9.8.7.6}}{\mathrm{1.2.3.4}}$
otherwise also ${{}^{8}C}_3+{{}^{8}C}_4={{}^{9}C}_4$
Hence total $=28+56+70=28+126=154$ ways.