If P r + 6 56 : 54 P r + 3 = 30800 : 1 , find r .

We know, P r n = n ! ( n - r ) ! P r + 6 56 : P r + 3 54 = 30800 : 1 ∴ 56 ! ( 56 - r - 6 ) ! 54 ! ( 54 - r - 3 ) ! = 30800 1 ⇒ 56 ! × ( 51 - r ) ! ( 50 - r ) ! × 54 ! = 30800 ⇒ 56 × 55 × 54 ! × ( 50 - r ) ! × ( 51 - r ) ( 50 - r ) ! × 54 ! = 30800 or simplifying, we get ( 51 - r ) = 10 ∴ r = 41

If C 2 r 28 : C 2 r - 4 24 = 225 : 11 , find r .

We know C r n = n ! ( n - r ) ! · r ! We are given C 2 r 28 : C 2 r - 4 24 = 225 : 11 ∴ C 2 r 28 C 2 r - 4 24 = 225 11 ∴ 28 ! ( 28 - 2 r ) ! · ( 2 r ) ! 24 ! ( 24 - 2 r + 4 ) ! · ( 2 r - 4 ) ! = 225 11 or ( 28 ) ! × ( 28 - 2 r ) ! ( 2 r - 4 ) ! ( 2 r ) ! × ( 28 - 2 r ) ! × 24 ! = 225 11 or 28 × 27 × 26 × 25 2 r ( 2 r - 1 ) ( 2 r - 2 ) ( 2 r - 3 ) = 225 11 or 2 r ( 2 r - 1 ) ( 2 r - 2 ) ( 2 r - 3 ) = 14 × 13 × 12 × 11 ∴ r = 7 .

E M B I B E

Higher Algebra>Chapter 11 - Permutations and Combinations>EXAMPLES. XI. a.>Q 27

HARD

JEE Main

IMPORTANT

Earn 100

A boat is to be manned by eight men, of whom $2$ can only row on bow side and one can only row on stroke side; in how many ways can the crew be arranged?

Important Questions on Permutations and Combinations

MEDIUM

JEE Main

IMPORTANT

Higher Algebra>Chapter 11 - Permutations and Combinations>EXAMPLES. XI. a.>Q 28

There are two works, each of three volumes and two works each of two volumes, in how many ways can the books be placed on a shelf, so that the volumes of the same work are not separated?

MEDIUM

JEE Main

IMPORTANT

Higher Algebra>Chapter 11 - Permutations and Combinations>EXAMPLES. XI. a.>Q 29

In how many ways can

10

examination papers be arranged so that the best and the worst papers never come together?

HARD

JEE Main

IMPORTANT

Higher Algebra>Chapter 11 - Permutations and Combinations>EXAMPLES. XI. a.>Q 30

An eight-oared boat is to be manned by a crew chosen from

11

men, of whom

3

can steer but cannot row, and the rest can row but can not steer. In how many ways can the crew be arranged, if two of the man only row on the bow side?

EASY

JEE Main

IMPORTANT

Higher Algebra>Chapter 11 - Permutations and Combinations>EXAMPLES. XI. a.>Q 31

Prove that the number of ways in which