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Three Carnot engines operate in series between a heat source at a temperature T_1 = 600 K and heat sink at temperature T_4 = 100K (see figure). There are two other reservoirs at temperature T_2 " and " T_3 as shown, with T_1 gt T_2 gt T_3 gt T_4 The three engines are equally efficient if:

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Subtract : 3t ^(4) - 4t ^(3) + 2t ^(2) - 6t + 6 from - 4t ^(4) + 8t ^(3) - 4t ^(2) - 2t + 11

There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of class 10 and 11 is 100 k, then k is equal to _______.