Question

In how many ways first and second rank in Mathematics, first and second rank in Physics, first rank in Chemistry and first rank in English be given away to a class of 30 students.

Answer 1

Hint: Find the categories which are present in the given question. Find the possibilities present in each case. Now check whether you have to add the categories or multiply them to get the result of total possibilities. Here each subject ranks are independent choices. So, you must multiply all rank’s possibilities for each rank to be secured.

Complete step-by-step answer:
Rule of Sum: - In combinatorics, the rule of sum or addition principle is basic counting principle. It is simply defined as, if there are A ways of doing P work and B ways of doing Q work. P, Q words cannot be done together. Total number of ways to do both P, Q are given by (A + B) ways.
Rule of product: - In combinatorics, the rule of product or multiplication principle is basic counting principle. It is simply defined as, if there are A ways of doing P work and B ways of doing Q work. Given P, Q works can be done at a time. Total number of ways to do both P, Q works are given by (A.B) ways.
By listing all possible categories, we get:
Rank 1 in mathematics, Rank 2 in mathematics, Rank in 1 Physics, Rank 2 in Physics, Rank 1 in chemistry, Rank 1 in English.
Total number of students is 30. So, if Rank 1 is given to 1 student for Rank 2 we will have 29 students left because 1 student is already assigned to Rank 1.
By listing each category with their possibilities, we get:
Possibilities for a student to secure rank 1 in Mathematics = 30
Possibilities for a student to secure rank 2 in Mathematics = 29
Possibilities for a student to secure rank 1 in Physics = 30
Possibilities for a student to secure rank 2 in Physics = 29
Possibilities for a student to secure rank 1 in Chemistry = 30
Possibilities for a student to secure rank 1 in English = 30
If a student got Rank 1 in mathematics there is no rule to stop him securing Rank 1 in Physics.
So, similarly it can be said for any pair of subjects. So, we must use product rules.
By applying product rule here, we get:
Total possibilities = \[\left( 30\times 29 \right)\times \left( 30\times 29 \right)\times 30\times 30\]
By simplifying, we get the value as:
Total possibilities = 681, 210, 000.
Therefore, doing given work we have 681, 210, 000 ways.

Note: Be careful while writing possibilities for Rank 2 you must subtract the Rank 1 student. Don’t forget this point. Be careful while categorizing into product rule or sum rule. You must carefully check whether there is any dependence. Students generally use sum rules here but it is wrong. You must follow the product rule.