Gujarat Board GSEB Textbook Solutions Class 11 Maths Chapter 7 Permutations and Combinations Ex 7.1 Textbook Questions and Answers.

## Gujarat Board Textbook Solutions Class 11 Maths Chapter 7 Permutations and Combinations Ex 7.1

Question 1.

How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5, assuming.

(i) repetition of digits is allowed?

(ii) repetition of digits is not allowed?

Solution:

(i) There are five digits, viz. 1, 2, 3, 4 and 5.

Every digit can be selected any number of times.

Hence, we can select first digit 5 times.

The second digit 5 times and the third digit 5 times.

Hence, the number of ways in which the selection of three digits can be made = 5 Ã— 5 Ã— 5 ways = 125 ways.

(ii) Under the restriction, first digit can be selected in 5 ways. After the selection of first digit, four digits are left. Second digit can be selected in 4 ways and third digit can be selected in 3 ways.

The total number of ways of selecting three digits is 5 Ã— 4 Ã— 3 = 60.

Question 2.

How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6, if the digits can be repeated?

Solution:

Let 2 be fixed at unitâ€™s place. The tenâ€™s place can be filled up in 6 ways. The hundredâ€™s place can also be filled in 6 ways.

Number of numbers that can be formed when 2 is at unitâ€™s place = 6 Ã— 6 = 36.

Similarly, when 4 is at unitâ€™s place, the number of numbers that can be formed = 36.

Again, when 6 is at the unit’s place, the number of numbers that can be formed = 36.

âˆ´ The total numbers of ways when 3-digits numbers can be formed, the digits being repeated = 36 Ã— 3 = 108.

Question 3.

How many 4-letter code words are possible, using the first 10 letters of the English alphabet, if no letter can be repeated?

Solution:

First letter of the code word can be selected in 10 ways. After the selection of the first letter, we have 9 letters. Hence, the second letter can be selected in 9 ways. According to FPC, first two letters can be selected in 10 Ã— 9 ways.

Similarly, third letter can be selected in 8 way s and the fourth letter can be selected in 7 ways. According to FPC, first two letter can be selected in 7 ways. According to FPC, number of ways of selecting four letters out of 10 letters of English alphabet

= 10 Ã— 9 Ã— 8 Ã— 7 ways

= 5040 ways.

Question 4.

How many 5 – digit telephone numbers can be constructed using the digits 0 to 9, if each number starts with 67 and no digit appears more than once?

Solution:

There are 10 digits from 0 to 9.

I place after 67 can be filled in 8 ways.

II place after 67 can be filled in 7 ways.

III place after 67 can be filled in 6 ways.

âˆ´ No. of telephone numbers that can be constructed

= 8 Ã— 7 Ã— 6

= 336.

Question 5.

A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?

Solution:

In tossing a coin, there are two possible outcomes, viz. H or T (i.e., Head or Tail). Similarly, in tossing the coin for the second time, there are again the same two outcomes and similarly for third toss, the same outcomes appear.

âˆ´ Total number of outcomes in three tosses (according to FPC)

= 2 Ã— 2 Ã— 2 ways

= 23 ways

= 8 ways.

Question 6.

Given 5 flags of different colours, how many different signals can be generated, if each signal requires the use of 2 flags, one below the other?

Solution:

First place can be filled up in 5 ways. After filling fit place, we are left with 4 flags.

âˆ´ The second place can be filled up in 4 ways.

The total number of signals w’hich can be generated

= 5 Ã— 4

= 20.