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11th Maths Weekly Test-1 ( Combinatorics and Mathematical Induction )

St. Britto Hr. Sec. School - Madurai

11th Maths Weekly Test-1 ( Combinatorics and Mathematical Induction )-Aug 2020

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

If a^2-a C₂=a^2-a C₄ then the value of 'a' is

2
3
4
5

The number of integers greater than 6000 that can be formed, using the digits 3, 5, 6, 7 and 8 without repetition

216
192
120
72

The number of five digit telephone numbers having at least one of their digits repeated is

90000
10000
30240
69760

The number of 10 digit number that can be written by using the digits 2 and 3 is

¹⁰C₂+⁹C₂
2¹⁰
2¹⁰-2
10!

There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two points is

45
40
39
38

The number of rectangles than can be formed on a chess board is

⁹C₂
⁹C₂ X ⁹C₂
204
224

If ⁿP_t = 720 nCr, then the value of r =

6
5
4
7

There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two of them is

45
40
39
38

In a plane there are 10 points are there out of which 4 points are collinear, then the number of triangles formed is

110
¹⁰C₃
120
116

For all n \(\in \) N, 3\(\times\) 5²ⁿ⁺¹+2³ⁿ⁺¹ is divisible by

19
17
23
25

Part - B

Generic 2 - Answer All Question

If ⁵P_r = ⁷P_r-1 find r.

Find the total number of subsets of a set with [Hint: ⁿC₀+ⁿC₁+ⁿC₂+...+ⁿC_n=2ⁿ].
5 elements

How many triangles can be formed by 15 points, in which 7 of them lie on one line and the remaining 8 on another parallel line?

Prove that \(2n{ C }_{ n }=\frac { { 2 }^{ n }\times 13\times ...(2n-1) }{ n! } \) .

Part - C

Generic 3 - Answer All Question

Find the number of strings that can be made using all letters of the word THING. If these words are written as in a dictionary, what will be the 85^th string?

How many different strings can be formed together using the letters of the word "EQUATION" so that (i) the vowels always come together? (ii) the vowels never come together?

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: almost 3 girls?

Five boys and 5 girls form a line. Find the number of ways of making the seating arrangement under the following condition.

C₁ C₂

(a) Boys and girls sit alternate (i) 5! x 6!

(b) No two girls sit together (ii) 10! - 5! 6!

(c) All the girls sit together (iii) (5 !)² + (5!)²

(d) All the girls are never together (iv) 2! 5! 5!

Part - D

Generic 5 - Answer All Question

How many three-digit numbers are there with 3 in the unit place?
(i) with repetition
(ii) without repetition.

There are 11 points in a plane. No three of these lies in the same straight line except 4 points, which are collinear. Find,
(i) the number of straight lines that can be obtained from the pairs of these points?
(ii) the number of triangles that can be formed for which the points are their vertices?

Using the mathematical induction, show that for any natural number n > 2,
\(\left(1-{1\over 2^2} \right)\left(1-{1\over 3^2} \right)\left(1-{1\over 4^2} \right)...\left(1-{1\over n^2} \right)={n+1\over 2n}\)

A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be done if atleast 5 women have to be included in a committee? In how many of these committees the women are in majority?

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