There were 35 students in a hostel. If the number of students increased by 7, the expenses of the mess were increase by Rs 42 per Day , while the average expenditure per head diminished by Re. 1 The original expenditure of the mess was?

Let the number be 4N, 5N and 6N
Then, 4N + 5N + 6N/3 = 25
or 15N = 75
or N = 5
Then largest number = 6N = 6 x 5 = 30

There are 13 hearts and 3 more kings
? P(heart or a king) = (13 + 3)/52 = 4/13
? P (neither a heart nor a king) = 1 - 4/13 = 9/13

Total number of caps = 12
? n(S) = ¹²C₃ = 220
n(E₁) = Out of 4 red caps, number of ways to pick 2 caps = ⁴C₂ = 6
n(E₂) = Out of 5 green caps
Number of ways to pick one cap = ⁵C₁ = 5
P(E) = n(E₁) x n(E₂)/n(S) = (6 x 5)/220 = 3/22

Total Number of outcomes = ¹⁰C₃ = 120
Number of outcomes not containing red balls = ⁵C₃ = 10
? Probability that at least one is red = 1 - 10/120 = 11/12

Let length of bridge be L m.
We know, Speed = Distance/Time

According to the question,
90 x 5/18 = 150 + L/26
? 25 x 26 = 150 + L
? 650 = 150 + L
? L = 500 m

Total number of caps = 2 + 4 + 5 + 1 = 12
Total number of outcomes = n(S) = ¹² C₂ = 66
Favorable number of outcomes = n(E) = ²C₂ = 1
? Required probability = 1/66

Ways of selection of two green marbles = ²C₂ = 1
Ways of selection of two yellow marbles = ³C₂ = 3
So, probability (both are green) = 1/¹⁵C₂ ....(I)
Probability (both are yellow) = 3/¹⁵C₂ ....(Ii)

Then, required probability = 1/¹⁵C₂ + 3/¹⁵C₂ = 4/ ¹⁵C₂
= 4/105

? = 3/5 of 4/7 of 7/9 of 425
? ? = 3/5 x 4/7 x 7/9 x 425 = 113.33 = 113

? Loss% = (20/10)² = (2)² = 4%
? Total S.P. = Rs. 24000
? Total C.P. = Rs. (100/96) x 24000 = Rs. 25000
? Loss = Rs. 1000

Relative speed of policeman with thief = (10 - 8) = 2 km/h
Now, time taken to cover 100 m with speed of 2 km/h or 2000 m/h

Now, Speed = Distance / Time
? Time = D/S = 100/2000
= 1/20 h or 60/20 min = 3 min