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 ___ ___ ___ ___

1^st 2^nd 3^rd 4^th 5^th

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

2^nd dash --> We have 4 options

3^rd dash --> We have 3 options

4^rd dash --> We have 2 options

5^rd dash --> We have 1 option

= 1 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1

= 24 words

Case 2 :

Number of words ends with S,

___ ___ ___ ___ S

1^st 2^nd 3^rd 4^th 5^th

5^rd dash --> We have 1 option (S)

1^st dash --> We have 4 options

2^nd dash --> We have 3 options

3^rd dash --> We have 2 options

4^rd dash --> We have 1 options

= 4 ⋅ 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.

1^st question = 4 ways

2^nd 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

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HOW MANY STRINGS CAN BE FORMED WITH THE GIVEN WORD