Question

Five separate awards (best scholarship, best leadership qualities, and so on) are to be presented to selected students

from a class of $30$. How many different outcomes are possible if

(a) a student can receive any number of awards?

(b) each student can receive at most $1$award?

Short answer

(a) Number of possible outcomes are $24,30,000$.

(b) Number of possible outcomes are$17,100,720$.

Step-by-step solution

Part (a) Step 1. Possible number of outcomes if a student can receive any number of awards.

No. of students = $30$

No. of awards = $5$

It is given that a student can receives any number of awards, it means all students are eligible for all five awards.

Therefore, number of possible outcomes =$30\times 30\times 30\times 30\times 30=24,30,000$

Part (b) Step 2. Possible number of outcomes if a student can receive at most one award.

No. of students = $30$

No. of awards = $5$

As one student can receive at most one award, so

No. of students eligible for first award = $30$

No. of students eligible for second award = $29$

No. of students eligible for third award = $28$

No. of students eligible for fourth award = $27$

No. of students eligible for fifth award = $26$

Therefore, total no. of outcomes are$=30\times 29\times 28\times 27\times 26=17,100,720$