CoCobes Permutation and Combination Quiz-1

Total given digits = 5
We have to place these 5 digits at unit, tens, 100th, 1000th and 10000th place.
We cannot take 'O' at 1000th place, so this place can be filled in 4 ways.
Repetition of digits is allowed.
∴
Each place out of unit, 10th, 100th and 1000th can be filled in 5 ways.
Now form multiplication rule
Total numbers =4×5×5×5×5=2500

Number of ways = ¹⁰C₄ x ⁵C₂ = 210 x 10 = 2100

Total attempts possible with three rings = 5\times 5 \times\times 5 \times 5\times 5 = 3125

Out of 3125 attempts only 1 will be a successful attempt.
Therefore, total number of failed attempts  = 3125-1= 3124

Each of the 10 mangoes can be placed in any of the 5 boxes, that is each mangoes has possible 5 arrangements.

First mango can be arranged in 5 ways.

Second can be arranged in 5 ways.

There are total 8 alphabets in the “STRAIGHT” out of which only 2 are Vowels (A and I)

Number of consonants = 6

Let the position of letter be:

(1) (2) (3) (4) (5) (6) (7) (8)

Vowels can occupy only odd places = 1,3,5, and 7

Arrangement of vowels can be done in = ⁴P₂ ways = 12

Arrangement of consonants = 6! = 720

Total number of words = 12\times 720 =8640

Number of ways of selecting (3 consonants out of 7) and (2 vowels out of 4)
= (^{7}\textrm{C}_{3}\times^{4}\textrm{C}_{2})
= 210.
Number of groups, each having 3 consonants and 2 vowels = 210.
Each group contains 5 letters.
Number of ways of arranging
5 letters among themselves = 5!
= 5 x 4 x 3 x 2 x 1
= 120.
Required number of ways = (210 x 120) = 25200.

physics book can be arranged in 5!=120
both the book can be arranged in 2!=2 eg (MMM)(PPPPP) or (PPPPP)(MMM)
so total arrangement will be 2!\times 5!\times 3! = 2\times 120\times 6 = 1440

Number of ways of selecting 3 points from given 12 points = ¹²C₃
But any three points selected from given seven collinear points does not form triangle.
Number of ways of selecting three points from seven collinear points = ⁷C₃ Required number of triangles = ¹²C₃ – ⁷C₃ = 220 -35 = 185 .

There are 5 letters each and Each letter is unique.

No of arrangements = 5! = 120

^{6}\textrm{P}_{2}

= 6\times 5

= 30