Math IOQM

A country has a food deficit of 10 % . Its population grows continuously at a rate of 3 % per year. Its annual food production every year is 4% more than that of the last year. Assuming that the average food requirement per person remains constant, prove that the country will become self-sufficient in food after n years, where n is the smallest integer bigger than or equal to (ln 10 - ln 9)/ (ln ( 1.04 ) - 0.03). Solution: Key point to note here is that population is growing continuously whereas food production is growing annually. So P_n = P_0.e^(rt) F_n = F_0.(1 + r/100)^t

Question: Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen? Solution 1: We start by noting that each of the 4 boxes contains 3 red and 2 blue distinct balls. In any valid selection of 10 balls, from each box at least one red and one blue must be chosen. Step 1. Determine How Many Balls Are Chosen from Each Box Let x i x_i be the number of balls chosen from box i i (for i = 1 , 2 , 3 , 4 i=1,2,3,4 ). We have: x 1 + x 2 + x 3 + x 4 = 10 x_1+x_2+x_3+x_4 = 10 Since from each box at least one red and one blue must be chosen, we must choose at least 2 balls from each box: x i ≥ 2 for  i = 1 , 2 , 3 , 4. x_i \ge 2 \quad \text{for } i=1,2,3,4. Define y i = x i − 2 y_i = x_i - 2 (so y i ≥ 0 y_i \ge 0 ). Then: y 1 + y 2 + y 3 + y 4 = 10 − 8 = 2. y_1+y_2+y_3+y_4 = 10 - 8 = 2. The n...

Answer: 1