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How many 3 letter words can be formed from word PRACTICES?

P R A C T I C E S

This word contains 8 distinct letters ( P, R, A, C, T, I, E, S), among them C is repeated twice (2).

The qs. is to make 3-letter words out of this letter.

So, we can form 3-letter words in this ways:

1.  All the letters of the word (3-letter word) will be Distinct.

OR

2. The word will be having a letter which is repeated Twice.

So, the 1st one:

There are 8 distinct letters, out of which we have to choose 3 letters and arrange them.

Hence,  8P3 = 8!/(8-3)! = 8!/5! = 336

2nd:

So, we're having only 1 letter which is repeated Twice (1C1) & we have to select another letter from remaining 7 letters (7C1) & then we have to arrange the word (3!/2!).

(here 1 thing may comes into mind that why 3!/2! ? : the 3 -letter word is containing 3 letters, out of which 1 letter is distinct and another 2 letter is identical(alphabet 'C'), so 1 letter is repeated twice in a 3 letter word, therefore 3!/2!)

Hence,    1C1 7C1 * 3!/2! = 1 * 7!/{1!*(7-1)!} * 3 = 7*3 = 21

Total ways = 1st one OR 2nd one

= 336 + 21

= 357

answered by (5.7k points) 7 22 138
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$PRA'CC'TIES$
No of words with 0 repetition $: 8_{c_3}*3!=336$
No of words with 1 repetition $: 7*3=21 \\ Total : \color{Blue}{357}$
answered by (870 points) 2 16

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