SHOW WORK! SHOW STEPS! A professor has 21 students in her class. Each student wi

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SHOW WORK! SHOW STEPS!

A professor has 21 students in her class. Each student will receive a grade of A, B, C, D, or F. In how many ways might the professor assign a grade of A, B, C, D, or F to each of the 21 students in her class? The professor's university is compiling statistics on grade distributions and requests the grade distribution (i.e., the number of students in each grade category) from this class. How many grade distributions are possible for a class of 21 students? How many grade distributions are possible for a class of 21 students if the sets of A-students, B-students, C-students, D-students, and F-students form a partition of the class?

Explanation / Answer

(a) each of the student can get one of the 5 grades

So total number of ways of assigning the grades = 5*5*...*5(21 times) = 5^21

(b) Required number = number of nonnegative integer solutions of the equation

x1 + x2 +...+x5 = 21

(Her x1 denotes the number of students who got A,z2 no who got B and so on till x5 no of students who got F,total number of students = 21,since xi is the number of students it is non negative integer)

Thus required number = (21+5-1)C(5-1) = 25C4

(c)Here sets of A students,...F students dorm partition.

So xi all are nonzero,atleast one student should be there in each grade.

So required number = number of solutions of the equation x1+x2+...+x5 = 21 in positive integers = number of solutions of the equation y1+...+y5 = 16 , where yi = xi -1

= (16+5-1)C(5-1) = 20C4

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