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The number of four letter words that can be formed using the letters of the word BARRACK is
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The correct answer is:
270
270
If all four letters are different then the number of words ${ }^5 \mathrm{C}_4 \times 4 !=120$
If two letters are $\mathrm{R}$ and other two different letters are chosen from $\mathrm{B}, \mathrm{A}, \mathrm{C}, \mathrm{K}$ then the number of words $={ }^4 \mathrm{C}_2 \times \frac{4 !}{2 !}=72$
If two letters are $\mathrm{A}$ and other two different letters are chosen from $\mathrm{B}, \mathrm{R}, \mathrm{C}, \mathrm{K}$ then the number of words $={ }^4 \mathrm{C}_2 \times \frac{4 !}{2 !}=72$
If word is formed using two $R$ 's and two $A$ 's then the number of words $=\frac{4 !}{2 ! 2 !}=6$
Therefore, the number of four-letter words that can be formed $=120+72+72+6=270$
If two letters are $\mathrm{R}$ and other two different letters are chosen from $\mathrm{B}, \mathrm{A}, \mathrm{C}, \mathrm{K}$ then the number of words $={ }^4 \mathrm{C}_2 \times \frac{4 !}{2 !}=72$
If two letters are $\mathrm{A}$ and other two different letters are chosen from $\mathrm{B}, \mathrm{R}, \mathrm{C}, \mathrm{K}$ then the number of words $={ }^4 \mathrm{C}_2 \times \frac{4 !}{2 !}=72$
If word is formed using two $R$ 's and two $A$ 's then the number of words $=\frac{4 !}{2 ! 2 !}=6$
Therefore, the number of four-letter words that can be formed $=120+72+72+6=270$
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