Question

boy has 3 library tickets and 8 books of his interest in the library of these 8, he does not want to borrow

mathematics part II unless mathematics part-1 is also borrowed? In how many ways can he choose the

three books to be borrowed?​

Answer 1

$\large\underline{\sf{Solution-}}$

➢ Given that,

A boy has 3 library tickets and 8 books of his interest in the library.

Of these 8 books, he does not want to borrow Mathematics part II unless Mathematics part-1 is also borrowed.

So,

Two cases arises :-

➢ Case :- 1

When he borrow Mathematics Part- II.

Now, in this case, if he borrow Mathematics part - II, he too has to borrow Mathematics part - 1. So, he has to choose third book from the remaining 6 books.

So, Number of ways to pick 1 book, out of 6 is

$\rm \: = \: \: ^6C_1$

$\rm \: = \: \: \dfrac{6!}{1! \: (6 - 1)!}$

$\red{\bigg \{ \because \:^nC_r = \dfrac{n!}{r!(n -r )!} \bigg \}}$

$\rm \: = \: \: \dfrac{6!}{\:5!}$

$\rm \: = \: \: \dfrac{6 \times 5!}{\:5!}$

$\rm \: = \: \: 6 \: ways$

So, Number of ways in which 3 books can be borrowed in case 1 = 6 ways.

Case :- 2

➢ When he didn't borrow Mathematics part- II.

So, in this case he has to choose 3 books from remaining 7 books.

So, Number of ways of selecting 3 books out of 7 is

$\rm \: = \: \: ^7C_3$

$\rm \: = \: \: \dfrac{7!}{3! \: (7 - 3)!}$

$\red{\bigg \{ \because \:^nC_r = \dfrac{n!}{r!(n -r )!} \bigg \}}$

$\rm \: = \: \: \dfrac{7!}{3! \: 4!}$

$\rm \: = \: \: \dfrac{7 \times 6 \times 5 \times 4!}{3 \times 2 \times 1 \times \: 4!}$

$\rm \: = \: \: 35 \: ways.$

So, Number of ways in which 3 books can be borrowed in case - 2 = 35 ways.

Hence,

Number of ways in which he can borrow 3 books is

$\rm \: = \: \: 35 + 6$

$\rm \: = \: \: 41 \: ways$

Additional Information :-

$\boxed{ \sf{ \: ^{n}C_{1} = ^{n}C_{n - 1} = n}}$

$\boxed{ \sf{ \: ^{n}C_{0} = ^{n}C_{n} = 1}}$

$\boxed{ \sf{ \: ^{n}C_{r} = \frac{n}{r} \: \: ^{n - 1}C_{r - 1}}}$

$\boxed{ \sf{ \: ^{n}C_{r} + \: ^{n}C_{r - 1} = \: ^{n + 1}C_{r}}}$

$\boxed{ \sf{ \: \frac{^{n}C_{r}}{^{n}C_{r - 1}} = \frac{n - r + 1}{r} }}$