HOW MANY STRINGS CAN BE FORMED WITH THE GIVEN WORD

How Many Strings Can Be Formed With The Given Word :

Here we are going to see some practice questions based on  the concept fundamental principle of counting.

How many strings can be formed with the given word - Questions

Question 1 :

How many strings can be formed using the letters of the word LOTUS if the word (i) either starts with L or ends with S? (ii) neither starts with L nor ends with S?

Solution :

(i) either starts with L or ends with S? 

Case 1 :

Number of words starts with L,

  L   ___   ___   ___   ___

                        1st     2nd    3rd     4th    5th

1st dash --> We have 1 option (L)

2nd dash --> We have 4 options

3rd dash --> We have 3 options

4rd dash --> We have 2 options

5rd dash --> We have 1 option

=  1 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 

=  24 words

Case 2 :

Number of words ends with S,

 ___   ___   ___   ___   S

                        1st     2nd    3rd     4th    5th

5rd dash --> We have 1 option (S)

1st dash --> We have 4 options

2nd dash --> We have 3 options

3rd dash --> We have 2 options

4rd dash --> We have 1 options

⋅ 3 ⋅ 2 ⋅ 1 ⋅ 1

=  24 words

Case 3 :

Number of words starts with L and ends with S.

 L  3 ⋅ 2 ⋅ 1 ⋅ S

=  6

Either starts with L or ends with S  =  24 + 24 - 6

=  42 words

 (ii) neither starts with L nor ends with S?

Total number of words formed without restriction  =  5!

  =  120 words

Number of words starts with L nor S 

=  Total number of words -  Number of words either starts with L or ends with S

=  120 - 42

=  78 words

Question 3 :

(i) Count the total number of ways of answering 6 objective type questions, each question having 4 choices.

(ii) In how many ways 10 pigeons can be placed in 3 different pigeon holes ?

(iii) Find the number of ways of distributing 12 distinct prizes to 10 students?

Solution :

(i)  Each question is having 4 options.There are 4 ways to answer each question.

1st question  =  4 ways

2nd question  =  4 ways  .........

Total number of ways  =  46

(ii) In how many ways 10 pigeons can be placed in 3 different pigeon holes ?

Note : If n different objects are to be placed in m places, then the number of ways of placing is mn.

10  -  Number of pigeons (different things)

3  -  Number of places (pigeon holes)

Applying the formula mn,

Total number of ways  =  310

(iii)  12  -  Number of prizes (different things)

10  -  Number of place (students)

Applying the formula mn,

Total number of ways  =  1012

After having gone through the stuff given above, we hope that the students would have understood "How Many Strings Can Be Formed With the Given Word"

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