Maths Permutations and Combinations Test.8

Explanation:

This type of question becomes very easy when we assume three colour are red(R) blue(B) and Green(G).

We can choose any colour.

Now according to the statement 1 i.e.., codes can be generated by waving single flag of different colours, then number of ways are three i.e.., R.B.G from statement III three flags in different sequence of colours, then number of ways are six i.e.., RBG, BGR, GBR, RGB, BRG, GRB.

Hence total number of ways by changing flag = 3+ 6 +6 = 15

Explanation:

Given that, the question paper consists of five problems. For each problem, one or two or three or none of the choices can be attempted.

Hence, the required number of ways = 45 – 1.
= 210 – 1 = 1024 – 1 = 1023

Explanation:

We can select one boy from 20 boys in 20 ways.

We select one girl from 25 girls in 25 ways
We select a boy and girl in 20 * 25 ways i.e., = 500 ways.

Explanation:

The word contains five consonants. Three vowels, three consonants can be selected from five consonants in ⁵C₃ ways, two vowels can be selected from three vowels in ³C₂ ways.

3 consonants and 2 vowels can be selected in ⁵C₂ . ³C₂ ways i.e., 10 * 3 = 30 ways.

Explanation:

Total number of balls = 9 + 3 + 4

Two balls can be drawn from 16 balls in ¹⁶C₂ ways.

Explanation:

The given digits are 1, 2, 3, 5, 7, 9

A number is even when its units digit is even. Of the given digits, two is the only even digit.
Units place is filled with only ‘2’ and the remaining three places can be filled in ⁵P₃ ways.
Number of even numbers = ⁵P₃ = 60.

Explanation:

The given digits are six.

The number of four digit numbers that can be formed using six digits is ⁶P₄ = 6 * 5 * 4 * 3 = 360.

Explanation:

We can initially arrange the six boys in 6! ways.

Having done this, now three are seven places and six girls to be arranged. This can be done in ⁷P₆ ways.
Hence required number of ways = 6! * ⁷P₆

Explanation:

Since each ring consists of six different letters, the total number of attempts possible with the three rings is = 6 * 6 * 6 = 216. Of these attempts, one of them is a successful attempt.

Maximum number of unsuccessful attempts = 216 – 1 = 215.

Explanation:

Total number of persons in the committee = 5 + 6 = 11

Number of ways of selecting group of eight persons = ¹¹C₈ = ¹¹C₃ = (11 * 10 * 9)/(3 * 2) = 165 ways.

Explanation:

The number of ways to select three men and two women such that one man and one woman are always selected = Number of ways selecting two men and one woman from men and five women

= ⁴C₂ * ⁵C₁ = (4 * 3)/(2 * 1) * 5
= 30 ways.

Explanation:

The number of ways to select two men and three women = ⁵C₂ * ⁶C₃

= (5 *4 )/(2 * 1) * (6 * 5 * 4)/(3 * 2)
= 200

Explanation:

n items of which p are alike of one kind, q alike of the other, r alike of another kind and the remaining are distinct can be arranged in a row in n!/p!q!r! ways.

The letter pattern ‘MESMERISE’ consists of 10 letters of which there are 2M’s, 3E’s, 2S’s and 1I and 1R.
Number of arrangements = 9!/(2!)² 3!

Explanation:

The word MEADOWS has 7 letters of which 3 are vowels.

-V-V-V-
As the vowels have to occupy even places, they can be arranged in the 3 even places in 3! i.e., 6 ways. While the consonants can be arranged among themselves in the remaining 4 places in 4! i.e., 24 ways.
Hence the total ways are 24 * 6 = 144.

Explanation:

There are five letters in the given word.

Consider 5 blanks ….
The first blank and last blank must be filled with N and A all the remaining three blanks can be filled with the remaining 3 letters in 3! ways.
The number of words = 3! = 6.

Explanation:

Total number of letters = 8

Using these letters the number of 8 letters words formed is ⁸P₈ = 8!.

Explanation:

The number of letters in the given word is four.

The number of three letter words that can be formed using these four letters is ⁴P₃ = 4 * 3 * 2 = 24.

Explanation:

The boy can select one trouser in nine ways.

The boy can select one shirt in 12 ways.
The number of ways in which he can select one trouser and one shirt is 9 * 12 = 108 ways.