Question
How many different words can be formed by jumbling the letters in the word $$MISSISSIPPI$$ in which no two $$S$$ are adjacent?
A.
$$8.{\,^6}{C_4}.{\,^7}{C_4}$$
B.
$$6.\,7.{\,^8}{C_4}$$
C.
$$6.\,8.{\,^7}{C_4}$$
D.
$$7.{\,^6}{C_4}.{\,^8}{C_4}$$
Answer :
$$7.{\,^6}{C_4}.{\,^8}{C_4}$$
Solution :
First let us arrange $$M, I, I, I, I, P, P$$
Which can be done in $$\frac{{7!}}{{4!2!}}$$ ways
$$\sqrt {{M}} \sqrt {{I}} \sqrt {{I}} \sqrt {{I}} \sqrt {{I}} \sqrt {{P}} \sqrt {{P}} $$
Now $$4S$$ can be kept at any of the ticked places in $$^8{C_4}$$ ways so that no two $$S$$ are adjacent.
Total required ways
$$\frac{{7!}}{{4!2!}}{\,^8}{C_4} = \frac{{7!}}{{4!2!}}{\,^8}{C_4} = 7 \times {\,^6}{C_4} \times {\,^8}{C_4}$$