Permutation and combination Help!!!

[anuu]

Hi Guys,

I came across this problem in tutorvista site.

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Find the number of words which have atleast one letter repeated.


I followed a different approach for solving and got a diiferent /wrong answer. Here's the appraoch that i followed.

Since 5 letters need to formed with atleast 1 word repeated, we have the following 3 cases:

CASE 1) xx123, where 1 word (XX) is repeated and 123 r the remaining 3 words

case 2) xxyy4, where 2 words r repeated (XX and YY) and 4 is the last word

CASE 3) xxxxx where all the word r repeated


CASE 1:

xx123

1st letter can be selected from 10 leeters in 10c1ways, since the second letter is repeated, there is only 1 way to do this. similarly , 3rd, 4th and 5th posotion can be filled in 9c1, 8c1 and 7c1 ways respectively.

so the total is : 10c1*1*9c1*8c1*7c1 ways ..further each of these words can be arranged in 5!/2! ways

So toatal is : 10c1* 1 * 9c1*8c1*7c1 *5!/2! = 302400

CASE 2:

xxyy4

No of five letter words that can be formed = 10c1 *1 * 9c1 *1 *8c1 ways..furhter teses can be arranged in 5!/2!2! ways

So total number of 5 letters that can be formed, where two letters r repeated r = 10c1 *1 * 9c1 *1 *8c1 *5!/2!2! = 21600 ways

CASE 3)

xxxxx

10c1*1*1*1*1 ways . further this can be arraged in 1 ways only
So total number of 5 letter word that can be formed where alll leters r repeated is 10


TOTAL = CASE 1+CASE2+CASE3

302400 +21600+10

Can somebody please tell me where have i gone wrong?

Thanks,
Anu

[manpsingh87]

Hi Guys,

I came across this problem in tutorvista site.

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Find the number of words which have atleast one letter repeated.


I followed a different approach for solving and got a diiferent /wrong answer. Here's the appraoch that i followed.

Since 5 letters need to formed with atleast 1 word repeated, we have the following 3 cases:

CASE 1) xx123, where 1 word (XX) is repeated and 123 r the remaining 3 words

case 2) xxyy4, where 2 words r repeated (XX and YY) and 4 is the last word

CASE 3) xxxxx where all the word r repeated


CASE 1:

xx123

1st letter can be selected from 10 leeters in 10c1ways, since the second letter is repeated, there is only 1 way to do this. similarly , 3rd, 4th and 5th posotion can be filled in 9c1, 8c1 and 7c1 ways respectively.

so the total is : 10c1*1*9c1*8c1*7c1 ways ..further each of these words can be arranged in 5!/2! ways

So toatal is : 10c1* 1 * 9c1*8c1*7c1 *5!/2! = 302400

CASE 2:

xxyy4

No of five letter words that can be formed = 10c1 *1 * 9c1 *1 *8c1 ways..furhter teses can be arranged in 5!/2!2! ways

So total number of 5 letters that can be formed, where two letters r repeated r = 10c1 *1 * 9c1 *1 *8c1 *5!/2!2! = 21600 ways

CASE 3)

xxxxx

10c1*1*1*1*1 ways . further this can be arraged in 1 ways only
So total number of 5 letter word that can be formed where alll leters r repeated is 10


TOTAL = CASE 1+CASE2+CASE3

302400 +21600+10

Can somebody please tell me where have i gone wrong?

Thanks,
Anu

well cases that are possible in this case are
1) 2 same 3 different. xxabc
2) 3 same 2 different. xxxab.
3) 4 same 1 different. xxxxa.
4) 5 same xxxxx.
5) 2 same 3 same but different than already selected. xxyyy.
6) 2 same 2 same but different than already selected, 1 different than already selected numbers, xxyya.

apart from this some cases might get while doing their permutation..!!!

well my question is why do you want to go for the longer method..?? i don't think its advisable to solve it by using this method because in a real exam like condition you might not be able to recall all the possible cases..!!!

well i also solved it by using the same method as used by tutorvista..,, i'll post that method for all other readers.. for their reference.. you can point out the portion of an answer solution that you find difficult in comprehending..!!!

Total number of 5 letter words that can be formed from given 10 letters = 10^5
now, Total number of words formed from given 10 letters in which all the words are different=10*9*8*7*6=30240;
hence required no. of ways= 10^5-30240=69760

[telberrak]

Waht about this solution

NB : A WORD THAT CONTAIN 2 REPEATED LETTERS IS STILL A SINLE WORD
and also, the maximum words of 5 letters you can make from 10 letters in 10 power 5 = 100000


10power(5) = 100000 is the number all the possible words of five letters formed from the 10 letters (every letter can be repeated from 2 times to 5 times)

10A5 = 30240 is the number of possible words of 5 letters that doesn't conain any letter repeated, the order is important (AAXYZ is diffrent from XAAYZ)

so, the number of combinations that contain AT LEAST one letter repeated (which means 1 repeated, 2 repeated ..., 5 repeated) is the all words minus the words that doesn't contain any letter repeated :

10A5 - 10C5 = 30240 - 30240 = 69760 words