Question

There are nine students (5 boys & 4 girls) in the class. In how many ways
(i) One student (either girl or boy) can be selected to represent the class.
(ii) A team of two students (one girl & one boy) can be selected.
(iii) Two medals can be distributed. (no one get both)
(iv) One prize for Maths, two prizes for Physics and three prizes for Chemistry can be distributed. (No student can get more than one prize in same subject & prizes are distinct)

Answer 1

Given, 4 girls & 5 boys (9 students) in class

(i) One student can be selected from class in ${}^9{C}_1$, ways

$=9$ ways [choosing 1 out of 9 students]

(ii) To select a team (1 boy & 1 girl)

Ways to select 1 boy ${=}^5{C}_1=5$

Ways to select 1 girl ${=}^4{C}_1=4$

Required no of ways $=5\times 4=20$

(iii) Two distinct medals can be distributed in

${=}^9{C}_2\times 2!=72$

${}^9{C}_2\to$ choosing 2 students out of 9 students

$2!\to$ mark divestment of medals [permutation]

(iv) Ways to distribute 1 prize for maths ${}^9{C}_1=9$

Ways to distribute 2 prizes for physics ${}^9{C}_2\times 2!=72$

Ways to distribute 3 prizes for to chemistry ${=}^9{C}_3\times 3!=504$

Total number of required ways to distribute according

to given conditions $=9\times 72\times 504$

$=326592$