Let D n represents derangement of ' n ' objects. If D n + 2 = a D n + 1 + b D n ∀ n ∈ N , then find b a

Number of ways in which n letters can be put in n corresponding envelopes such that no letters goes in right envelopes is denoted by D n . ⇒ D n = n ! 1 - 1 1 ! + 1 2 ! - 1 3 ! + . . . . . - 1 n 1 n ! In case of derangement D n of letter l 1 you have to choose any envelope among E 2 , E 3 , . . . . E n in C 1 n - 1 number of ways. ⇒ D n = C 1 n - 1 p + q Where, p is the derangement, if letter l 2 goes to envelope E 1 and q is the derangement, if letter l 2 goes to envelope other than E 1 . Hence, from the definition we can say if letter l 2 goes to envelop E 1 , then we have derangement of n - 2 envelop in D n - 2 ways. So, p = D n - 2 Similarly, q = D n - 1 So, D n = C 1 n - 1 D n - 2 + D n - 1 Hence, D n + 2 = C 1 n - 1 D n + D n + 1 So, a = b = C 1 n - 1 ∴ b a = 1

i Find exponent of 3 in 20 ! ii Find number of zeros at the end of 45 !

⇒ . is greatest integer function. i We know, the exponent of p in n ! is n p + n p 2 + n p 3 + . . . Exponent of 3 in 20 ! is 20 3 + 20 3 2 + 20 3 3 ⇒ 6 . 67 + 2 . 22 + 0 . 74 = 6 + 2 + 0 = 8 . ii We know, the number of zero at the end of n ! is n 5 + n 5 2 + n 5 3 + . . . Since 2 × 5 = 10 , and exponent of 2 will be greater than exponent of 5 . Hence, number of zeroes will be equal to exponent of 5 . The number of zeroes at the end of 45 ! is 45 5 + 45 5 2 ⇒ 9 + 1 . 8 = 9 + 1 = 10 .

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In how many ways it is possible to divide six identical green, six identical blue and six identical red among two persons such that each gets equal number of item?

Important Questions on Permutation and Combination

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 8 - Permutation and Combination>Exercise-1>Q 38

Find the number of solutions of

x + y + z + w = 20

under the following conditions:
(i)

x, y, z, w

are whole number
(ii)

x, y, z, w

are natural number
(iii)

x, y, z, w \in \{1, 2, 3, \dots \dots \dots, 10\}

(iv)

x, y, z, w

are odd natural number

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 8 - Permutation and Combination>Exercise-1>Q 39

A person has

4

distinct regular tetrahedron dice. The number printed on

4

faces of dice are

- 3, - 1,

1

and

3

. The person throws all the

4

dice. Find the total number of ways of getting sum of number appearing on the bottom face of dice equal to

0

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Alpha Question Bank for Engineering: Mathematics>Chapter 8 - Permutation and Combination>Exercise-1>Q 40

Five balls are to be placed in three boxes in how many different ways can be placed the balls so that no box remains empty if

(i)

balls and boxes are different

(ii)

balls are identical but boxes are different

(iii)

balls are different but boxes are identical

(iv)

balls as well as boxes are identical

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Alpha Question Bank for Engineering: Mathematics>Chapter 8 - Permutation and Combination>Exercise-1>Q 41

Let

D_{n}

represents derangement of

' n'

objects. If

D_{n + 2} = a D_{n + 1} + b D_{n} \forall n \in N,

then find

\frac{b}{a}

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Alpha Question Bank for Engineering: Mathematics>Chapter 8 - Permutation and Combination>Exercise-1>Q 42

A person writes letters to five friends and addresses on the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that
(a) all letters are in the wrong envelopes?
(b) at least three of them are in the wrong envelopes?

EASY

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 8 - Permutation and Combination>Exercise-1>Q 43

(i)

Find exponent of

3

20!

(ii)

Find number of zeros at the end of

45!

EASY

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 8 - Permutation and Combination>Exercise-1>Q 44

Find the total number of ways of selecting two numbers from the set of first

100

natural numbers such that difference of their square is divisible by

3

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 8 - Permutation and Combination>Exercise-1>Q 45

A four digit number plate of a car is said to be lucky if sum of first two digit is equal to sum of last two digit. Then find the total number of such lucky plate. (Assume

0000, 0011, 0111, \dots \dots \dots .

all are four digit number)

In how many ways it is possible to divide six identical green, six identical blue and six identical red among two persons such that each gets equal number of item?

Important Questions on Permutation and Combination

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