One type of liquid contains 25 % of benzene, the other contains 30% of benzene. A can is filled with 6 parts of the first liquid and 4 parts of the second liquid. Find the percentage of benzene in the new mixture.

If Mrs. X is selected among the ladies in the committee, then Mr. Y is not selected or if Mrs. X is not selected then Mr. Y can be there in the committee...
So, required number of ways
= ⁸C₃ x ⁶C₄ + ⁷C₃ x ⁷C₄
= [(8 x 7 x 6)/(3 x 2)] x [(6 x 5)/(2 x 1)] + [(7 x 6 x 5)/(3 x 2)] x [(7 x 6 x 5)/(3 x 2)]
= 840 + 1225
= 2065

T.D. ?(P.W. x B.G. )
= ?(576 x 1) = Rs. 24

First, we can select 12 beads out of 18 beads in ¹⁸C₁₂ ways.
Now, these 12 beads can make a necklace in 11! / 2 ways as clockwise and anti-clockwise arrangements are same.

So, required number of ways = [ ¹⁸C₁₂ .11! ] / 2!
= [¹⁸C₆.11! ] / 2!
= [18 x 17 x 16 x 15 x 14 x 13 x 11! ] / [6 x 5 x 4 x 3 x 2 x 1 x 2!]
= [17 x 7 x 13! ] / 2!
= 119 x 13 !/ 2

S_d = 15 km/h
S_u = 9 km/h
Speed of man= (S_d + S_u)/2
? Speed of man = (15 + 9)/ 2 = 12 km/h

Suppose the vessel initially contains 8 litres of liquid.

Let x litres of this liquid be replaced with water.
Quantity of water in new mixture =   3 - 3 x 8 + xlitres.

Quantity of syrup in new mixture =   5 - 5 x 8 litres.

   3 - 3 x 8 + x = 5 - 5 x 8

=>   5x + 24 = 40 - 5x 

=>   10x = 16    => x = 8/5

So, part of the mixture replaced =  8 5 × 1 8 = 1/5.

Let speed of the flow of water be x km/h and rate of sailing of sailor be y km/h.
Downstream speed (x + y) = 48/8
? x + y = 6 ....(i)
and upstream speed x - y = 4 ...(ii)
On solving Eqs. (i) and (ii), we get
y = 1 km/h

By the rule of alligation:
Cost of 1 kg rice of 1st kind                  Cost of 1 kg rice of 2nd kind 

  Required ratio = 60 : 90 = 2 : 3

Let initial quantity be Q, and final quantity be F

F = Q(1 - 8/Q) 4

=> Q = 20

Total share given by the person = (1/4) + (1/2) + (1/5)
= (5 + 10 + 4) / 20
=19/20

h = 8 m.

So, r = √l² - h² = √(10)² - 8² = 6 m.

∴ Curved surface area = Πrl = (Π x 6 x 10) m² = 60Π m².