C(n,k) represents the number of ways to choose k items from n, which corresponds to the coefficient of x^k.

The sum of the exponents in the term x^4y^3 is 4 + 3 = 7, hence n must be 7.

The number of terms n can be calculated using the formula: n = (last - first) / difference + 1. Here, n = (50 - 10) / 5 + 1 = 9.

The nth term of an arithmetic sequence is given by a + (n-1)d. Here, a = 5, d = 3, n = 10. So, 5 + (10-1)3 = 32.

Substituting n = 7 into the formula gives a_7 = 5*7 - 3 = 35 - 3 = 32.

The probability of failing is 1 - P(passing) = 1 - 0.75 = 0.25.

The probability of either A or B occurring is P(A) + P(B) - P(A and B) = 0.2 + 0.5 - (0.2 * 0.5) = 0.7.

For independent events, P(A and B) = P(A) * P(B) = 0.4 * 0.5 = 0.2.

For mutually exclusive events, P(C or D) = P(C) + P(D) = 0.2 + 0.3 = 0.5.

The probability that it will not rain is 1 - 0.7 = 0.3.

The constant ratio of consecutive terms in a geometric series is called the common ratio.

If all squares are rectangles, it implies that there are rectangles that are not squares, thus some rectangles are not squares.

The common difference can be found by calculating S_n - S_(n-1). Here, S_n = 3n^2 + 2n and S_(n-1) = 3(n-1)^2 + 2(n-1). The difference simplifies to 4.

'a' represents the first term of the arithmetic series.

The complement of set A consists of elements in the universal set U that are not in set A. Therefore, the complement is {1, 3, 5}.

The combinations that give a sum of 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), totaling 6 combinations. The total outcomes are 36, so the probability is 6/36 = 1/6.

The probability that both are smokers is 0.3 * 0.3 = 0.09.

The probability that it does not rain = 1 - P(rain) = 1 - 0.1 = 0.9.

The probability of losing all 5 games is (1 - 0.3)^5 = 0.168. Therefore, the probability of winning at least once is 1 - 0.168 = 0.832, which rounds to 0.836.

The relationship of direct proportionality is represented by the equation x = ky, where k is a constant.

To find the number of students who study only Mathematics, we use the formula: Only Mathematics = Total Mathematics - Both subjects. Thus, 18 - 10 = 8.

The expected number of students selected is calculated as 30 * 0.4 = 12.

To find the number of students who study only Hindi, we subtract those who study both from those who study Hindi: 25 - 10 = 15.

The number of students studying only English is 20 - 10 = 10, and only Mathematics is 25 - 10 = 15. Thus, total studying only one subject is 10 + 15 = 25.

The number of students who study only Science is 25 - 10 = 15.

Using inclusion-exclusion, the number of students who play either sport is: (Cricket + Football - Both) = 12 + 15 - 5 = 22.

The probability of losing all 4 games is (0.75)^4 = 0.3164. Therefore, the probability of winning at least once is 1 - 0.3164 = 0.6836, approximately 0.84.

The sum of the first n terms of a geometric series is a(1 - r^n) / (1 - r). Here, a = 4, r = 2, n = 5. So, 4(1 - 2^5) / (1 - 2) = 4(1 - 32) / -1 = 124.

The nth term of a geometric series is given by ar^(n-1). Here, a = 4, r = 2, n = 6. So, 4 * 2^(6-1) = 4 * 32 = 128.

The number of people who like at least one sport is 90 + 60 - 30 = 120, so those who like neither is 150 - 120 = 30.