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