Two digit number are formed from the digits 0, 1, 2, 3, 4 where digits are not repeated. Find the probability of the events that : (i) the number formed is an even number. (ii) the number formed is greater than 40. (iii) the number formed is prime number.

6. Two digit number are formed from the digits 0, 1, 2, 3, 4 where digits are not repeated. Find the probability of the events that :

(i) the number formed is an even number.

(ii) the number formed is greater than 40.

(iii) the number formed is prime number. 

Sol. The two digit numbers that can be formed using the digits 0, 1, 2, 3, 4 without repeating digits are

∴  S = { 10, 12, 13, 14, 20, 21, 23, 24 30, 31, 32, 34, 40, 41, 42, 43 }

∴  n (S) = 16

(i) Let A be the event that the number formed is even number

A = { 10, 12, 14, 20, 24, 30, 32, 34, 40, 42 }

n (A) = 10

P (A) = n (A)/n (S)

∴  P (A) = 10/16

∴  P (A) = 5/8

(ii) Let B be the event that number formed is greater than 40

B = { 41, 42, 43 }

n (B) = 3

P (B) = n (B)/n (S)

∴  P (B) = 3/16

(iii)Let C be the event that number formed is prime number

C = { 13, 23, 31, 41, 43 }

n (C) = 5

P (C) = n (C)/n (S)

∴  P (C) = 5/16

Form two digit numbers using the digits 0, 1, 2, 3, 4, 5 without repeating the digits. P is the event that the number so formed is even. Q is the event that the number so formed is divisible by 3. R is the event that the number so formed is greater than 50.

(v) Form two digit numbers using the digits 0, 1, 2, 3, 4, 5 without repeating the digits.

P is the event that the number so formed is even.

Q is the event that the number so formed is divisible by 3.

R is the event that the number so formed is greater than 50.

Sol. The two digit numbers that can be formed using the digits 0, 1, 2, 3, 4, 5 without repeating digits are as follows :

S = { 10, 12, 13, 14, 15, 20, 21, 23, 24, 25, 30, 31, 32, 34, 35, 40, 41, 42, 43, 45, 50, 51, 52, 53, 54 }

n (S) = 25

P is the event that number so formed is even

P = { 10, 12, 14, 20, 24, 30, 32, 34, 40, 42, 50, 52, 54 }

∴  n (P) = 13

Q is the event that number so formed is divisible by 3

Q = { 12, 15, 21, 24, 30, 42, 45, 51, 54 }

∴  n (Q) = 9

R is the event that the number so formed is greater than 50

R = { 51, 52, 53, 54 }

∴  n (R) = 4

4, 3, 2, 1, ....

(vii) 4, 3, 2, 1, .... (1 mark)

Sol. t1 = 4, t2 = 3, t3 = 2, t4 = 1,

t2 – t1 = 3 – 4 = – 1

t3 – t2 = 2 – 3 = – 1

t4 – t3 = 1 – 2 = – 1

Here, The difference between ANY two consecutive terms is –1 which is constant.

∴ The sequence is an A.P.

3, 5, 7, 9, 11, ..... (1 mark)

(ii) 3, 5, 7, 9, 11, ..... (1 mark)

Sol. t₁ = 3, t2 = 5, t3 = 7, t4 = 9, t5 = 11

t₂ – t₁ = 5 – 3 = 2

t₃ – t₂ = 7 – 5 = 2

t₄ – t₃ = 9 – 7 = 2

t₅ – t₄ = 11 – 9 = 2

Here, The difference between ANY two consecutive terms is 2 which is constant.

∴ The sequence is an A.P.

1, 3, 6, 10, ..... (1 mark)

(i) 1, 3, 6, 10, ..... (1 mark)

Sol. t₁ = 1, t₂ = 3, t₃ = 6, t4 = 10

t₂ – t₁ = 3 – 1 = 2

t₃ – t₂ = 6 – 3 = 3

t₄ – t₃ = 10 – 6 = 4

Here, The difference between ANY two consecutive terms is not constant.

∴  The sequence is not an A.P.

(iii) 1, 3, 7, 15, 31, .... (1 mark)

(iii) 1, 3, 7, 15, 31, .... (1 mark)

Sol. t₁ = 1

t₂ = t₁ + 2 = 1+ 2 = 3

t₃ = t₂ + 4 = 3 + 4 = 7

t₄ = t₃ + 8 = 7 + 8 = 15

t₅ = t₄ + 16  = 15 + 16 = 31

t₆ = t₅ + 32  = 31 + 32 = 63

t₇ = t₆  + 64  = 63 + 64 = 127

t₈ = t₇  + 128  = 127 + 128 = 255

t₉ = t₈  + 256  = 255 + 256 = 511

∴ The next four terms of the sequence are 63, 127, 255 and 511.

State whether the following sequence is an A.P. or not: 1, 3, 6, 10, ………

Ans.

Here, t1 = 1, t2 =3, t3 =6, t4 = 10

∴ t2 – t1 = 3 – 1 = 2

∴ t3 – t2 = 6 – 3 = 3

∴ t4 – t3 = 10 – 6 = 4

Here, the difference between any two   consecutive term is not constant

∴ it is not an A.P.

Mohan has a hobby of solving problems every day. He solved 2, 4, 3, 2, 4, 5, 1 problems in the last week. What is the mean number of problems solved by Mohan per day?

Solution:

Since Mohan solved 2, 4, 3, 2, 4, 5, 1 problems in the last week,

∴ Mean = Σx = 2 + 4 + 3 + 2 + 4 + 5 + 1 = 21 = 3 n 7 7

∴ Mean = 3

It means on an average Mohan solved 3 problems per day.

Two digit number are formed from the digits 0, 1, 2, 3, 4 where digits are not repeated. A is the event that the number formed is even. Write S, A , n(S) and n(A).

Solution:

Since two digit numbers are formed from the digits 0, 1, 2, 3, 4 where digits are not repeated.

∴ Sample Space S = {10, 12, 13, 14, 20, 21, 23, 24, 30, 31, 32, 34, 40, 41, 42, 43}

∴ n(S) = 16

Let A = Event that the number formed is even number.

∴ A = {10, 12, 14, 20, 24, 30, 32, 34, 40, 42}

∴ n(A) = 10