How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two…

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two $\mathrm{S}$ are adjacent?
  1. $8 \cdot{ }^6 \mathrm{C}_4 \cdot{ }^7 \mathrm{C}_4$
  2. $6.8 \cdot{ }^7 \mathrm{C}_4$
  3. $6 \cdot 7 \cdot{ }^8 \mathrm{C}_4$
  4. $7 \cdot{ }^6 \mathrm{C}_4 \cdot{ }^8 \mathrm{C}_4$

Solution

Other than $\mathrm{S}$, seven letters $\mathrm{M}, \mathrm{I}, \mathrm{I}, \mathrm{I}, \mathrm{P}, \mathrm{P}, \mathrm{I}$ can be arranged in $\frac{7 !}{2 ! 4 !}=7.5 .3$. Now four $\mathrm{S}$ can be placed in 8 spaces in ${ }^8 \mathrm{C}_4$ ways. Desired number of ways $=7.5 \cdot 3 \cdot{ }^8 \mathrm{C}_4=7 \cdot{ }^6 \mathrm{C}_4 \cdot{ }^8 \mathrm{C}_4$

Asked in: JEE Main 2008

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