basic aptitude

Question

Ten different letters of alphabet are given, words with 5 letters are formed from these given letters. Then, the number of words which have at least one letter repeated is:

	A.	69760
	B.	30240
	C.	99748
	D.	42386

i got ans from the method totol words - no word repeated , but if i want to do with general method means 10*10*9*8*7 *5!/2! + 10*10*10*9*8*5!/3! + 10*10*10*10*9*5!/4! + 10*10*10*10*!0   from this getting different ans where is going wrong ? someone verify pls 

Answer 1

In simple way answer can be calculate as number of words with atleast one letter repeated = total number of words from 10 letters - total number of words without repetition from 10 letters = (10^5) - (10P5) =100000 - (10*9*8*7*6)= 69760

Now as you need the answer using other way we can do the following:

No of ways of selecting with 1 letter repeated only once like A A _ _ _ =  10∗1∗9∗8∗7∗5!/3!∗2!

Following same procedure for 1 letter repeated twice,thrice and four times,

number of ways of repeating 1 letter = (10∗1∗9∗8∗7∗5!/(3!∗2!)) ONCE

+(10∗1∗1∗9∗8∗5!/(2!∗3!)) TWICE

+(10∗1∗1∗1∗9∗5!/(1!∗4!)) THRICE

+(10∗1∗1∗1∗1∗5!/5!) FOUR times

= 58060

Two letters repetition = $\frac{(10*9)}{2!}*8*\frac{5!}{2!*2!} + \frac{(10*9)}{2!}*2*\frac{5!}{3!*2!}$

First factor of sum is 2 letters where each repeated exactly twice in 5 letter word.

Second factor of sum is one of the letters is repeated more than twice and other letter letter is repeated exactly twice in 5 letter word.

which gives answer as 10800+900=11700

58060+11700 = 69760

Answer 2

No of ways of selecting with 1 letter repeated =  $10 * 1*9*8*7*5! /3!*2!$

Because after selecting a letter, to repeat it again ,there is only one way selecting same letter again.

Following same procedure,

no of ways = $10 * 1*9*8*7*5! /3!*2! + 10*1*1*9*8*5!/2!*3!+10*1*1*1*9*5!/1!*4!+10*1*1*1*1*5!/5!$

Comments

One letter repeated total = 58060... from above calculations

Two letters repetition = $\frac{(10*9)}{2!} *8*\frac{5!}{2!*2!} + \frac{(10*9)}{2!} *2*\frac{5!}{3!*2!}$

First factor of sum is Each letter repeated exactly twice in 5 letter word.

Second factor of sum is one of the letters is repeated more than twice in 5 letter word.

which gives answer as 10800+900=11700

58060+11700 = 69760.

Add this in the answer if u find no mistake in this @venkat_sirvisetti

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Are u considering cases when all letters are repeated and 4 letters are repeated like aaaaa, aaaab etc?

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Case aaaaa is already considered in ur answer in term 10*1*1*1*1*5!/5!

Case aaaab is also considered in ur answer in term 10*1*1*1*9*5!/(1!*4!)

Hence these cases are included in the count 58060