Question

a) For the production of a local play, 8 people auditioned and 6 of them joined the cast. Of these, 4 had named roles. In how many ways could this have happened?

b) In how many ways can 55 boys and 55 girls be arranged in a circle so that the boys and girls are alternating

c) How many 55-digit numbers have the property that the first and last digits are different?

Answer 1

a) Total Number of ways in which it can happen = (Total Number of ways of selecting 6 people from 8 ) * (Total Number of ways of selecting 4 people from selected 6 people)

Total number of ways it can happen

answer

b) In order to alternately arrange 55 boys and 55 girls, we would have to first make 55 boys sit in a circle with a gap between them. Since the boys are sitting in a circle hence the number of gaps between them will also be equal to 55.

Now, the total number of ways in which 55 boys can be arranged among themselves

Now, in the 55 gaps present between the boys, we will arrange 55 girls. These 55 girls can again be arranged among themselves in ways.

Now, for every arrangement of girls, ways of the arrangement of boys are possible. Hence, for number of the arrangement of girls, the total number of ways in which 55 boys and 55 girls can be arranged alternatively

(answer)

c) Suppose there is a 55 digit number that we want to write following the condition that the first and the last digits are different.

Now, the number of choices for the first digit = 10

Number of choices for the second digit = 10

similarly, no of choices from the second digit up to the 54th digit, will all be = 10

The number of choices for the 55th digit = 9 (Excluding the choice for the first digit since we want the number to have a different first and last digit.)

Hence, the total number of 55 digit numbers that have the given property

(answer)

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