Question

A bibliophile plans to put a total of seven books on her marble shelf. She can choose these seven books from a mixture of works from Antiquity and works on Post-modernism, of which there are seven each. If the shelf must contain at least four works from Antiquity, and one on Post-Modernism, then how many ways can he select seven books to go on the shelf?

Answer 1

Answer:

$1715$

Step-by-step explanation:

GIVEN: A bibliophile plans to put a total of seven books on her marble shelf. She can choose these seven books from a mixture of works from Antiquity and works on Post-modernism, of which there are seven each.

TO FIND: If the shelf must contain at least four works from Antiquity, and one on Post-Modernism, then how many ways can he select seven books to go on the shelf.

SOLUTION:

Total number of books available $=14$

As there must be at least $4$ books on  Antiquity and $1$ on Post-modernism

Therefore,

when there are $6$ Antiquity books and $1$ on Post-modernism.

total ways of selection $=^7C_6\times^7C_1=49$

when there are $5$ Antiquity books and $2$ on Post-modernism.

total ways of selection $=^7C_5\times^7C_2=21\times21$

                                                          $=441$

when there are $4$ Antiquity books on $3$ books on Post-modernism.

total ways of selection $=^7C4\times^7C3=35\times35$

                                                           $=1225$

Total ways of selection are $=1225+441+49$

                                             $=1715$

Hence the total ways of selection such that there are at least $4$ books on  Antiquity and $1$ on Post-modernism.