The ratio of the weights of two spheres is 8 : 27 and the ratio of weights per 1 cc of materials of two is 8 : 1 . The ratio of the radii of the spheres is

Work to be done by C = (1 - 7/11) = 4/11
? (A + B) : C = 7/11 : 4/11 = 7 : 4
? C's share = Rs. 550 x (4/11) = Rs. 200

There are 7 letters in the word Bengali of these 3 are vowels and 4 consonants.

There are 4 odd places and 3 even places. 3 vowels can occupy 4 odd places in 4 P 3 ways and 4 constants can be arranged in  4 P 4 ways.

Number of words = 4 P 3  x  4 P 4= 24 x 24 = 576

From 5 consonants, 3 consonants can be selected in  5 C 3 ways.

From 4 vowels, 2 vowels can be selected in  4 C 2 ways.

Now with every selection, number of ways of arranging 5 letters is 5 P 5ways.

Total number of words =  5 C 3 * 4 C 2 * 5 P 5

                                = 10x 6 x 5 x 4 x 3 x 2 x 1= 7200

Choose 1 person for the single room & from the remaining choose 2 for the double room & from the remaining choose 4 people for the four person room, 

 Then, ⁷C₁ x ⁶C₂ x ⁴C₄ 

_{= 7 x 15 x 1 = 105}

? Work of (Ram + Dilip + Shekhar) for 1 day = 1/20
and work of (Ram + Dilip) for 1 day = 1/30
? Work of Shekhar for 1 day = 1 / 20 - 1 / 30 = 1/60
Hence, Shekhar will complete the work in 60 days.

Number of words with 5 letters from given 9 alphabets formed =  9 5

Number of words with 5 letters from given 9 alphabets formed such that no letter is repeated is =  9 P 5

Number of words can be formed which have at least one letter repeated =   9 5 - 9 P 5

= 59049 - 15120

= 43929

ST candidates vacancies can be filled by C 3 5 ways = 10 

Remaining vacancies are 5 that are to be filled by 12 

=> C 5 12= (12x11x10x9x8)/(5x4x3x2x1) = 792 

Total number of filling the vacancies = 10 x 792 = 7920

Number of ways of choosing 2 black pens from 5 black pens in 5 C 2 ways.

Number of ways of choosing 2 white pens from 3 white pens in  3 C 2 ways.

Number of ways of choosing 2 red pens from 4 red pens in 4 C 2ways.

By the Counting Principle, 2 black pens, 2 white pens, and 2 red pens can be chosen in 10 x 3 x 6 =180 ways.

Time taken by Ramesh alone = (2/3) x 4 = 8/3 days
? Their 1 day's work = (1/4 + 1/6 + 3/8) = 19/24
So, together they can finish the work in 24 /19 days, i.e., 1⁵/₁₉ days.