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प्रश्न
Determine the number of arrangements of letters of the word ALGORITHM if consonants are at even positions.
उत्तर
In the word ALGORITHM even positions are 4 and consonants are 6.
∴ number of arrangements in which 6 consonants occupy 4 even positions = 6P4
Now, the remaining 5 positions can be occupied by remaining 5 letters in 5P5 ways.
Hence, the total number of words in which consonants occupy even positions
= 6P4 × 5P5
= `(6!)/((6 - 4)!) × 5! ...[(""^"n""P"_"r" = ("n"!)/(("n" - "r")!)),(""^"n""P"_"r" = "n"!)]`
= `(6!)/(2!) × 5!`
= `(6 × 5 × 4 × 3 × 2!)/(2!) × 5!`
= 6 × 5 × 4 × 3 × (5 × 4 × 3 × 2 × 1)
= 360 × 120
= 43200
∴ 43200 words can be formed if the consonants are at even positions.
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