Question 14 Marks
Out of 7 boys and 5 girls a team of 7 students is to be made.
(a) Find the number of ways, if team contain at least 3 girls.
(b) Find the number of ways, if team contain exactly 3 girls.
(c) if exactly 3 girls are selected and are arranged in a row for photograph. Find number of ways if all girls and all the boys will stand together.
(d) The number of ways to arrange 3 girls and 4 boys if no two boys and girls will stand together.
(a) Find the number of ways, if team contain at least 3 girls.
(b) Find the number of ways, if team contain exactly 3 girls.
(c) if exactly 3 girls are selected and are arranged in a row for photograph. Find number of ways if all girls and all the boys will stand together.
(d) The number of ways to arrange 3 girls and 4 boys if no two boys and girls will stand together.
Answer
View full question & answer→Read the text carefully and answer the questions:
Out of 7 boys and 5 girls a team of 7 students is to be made.
(i) 7 boys, 5 girls
ways to select at least 3 girls
$=3$ girls 4 boys or 4 girls 3 boys or 5 girls 2 boys
$={ }^{5} \mathrm{C}_{3} \times{ }^{7} \mathrm{C}_{4}+{ }^{5} \mathrm{C}_{4} \times{ }^{7} \mathrm{C}_{3}+{ }^{5} \mathrm{C}_{5} \times{ }^{7} \mathrm{C}_{2}$
$=10 \times 35+5 \times 35+1 \times 21$
$=350+175+21$
$=546$
(ii) Ways to select exactly three girls
$=3$ girls 4 boys
$={ }^{5} \mathrm{C}_{3} \times{ }^{7} \mathrm{C}_{4}=350$
(iii)Ways of arranging 3 girls and 4 boys if all girls and boys stand together
$=2!\times 3!\times 4!$
$=2 \times 6 \times 24$
$=288$
Total ways of selecting and arranging
$=288 \times 350$
$=100800$
(iv)Ways to arrange boys $=4$ !
B _ B _ B _ B
Ways to arrange girls $=3$ !
Total ways of selecting and arranging
$=4!\times 3!\times 350$
$=24 \times 6 \times 350$
$=50400$
Out of 7 boys and 5 girls a team of 7 students is to be made.
(i) 7 boys, 5 girls
ways to select at least 3 girls
$=3$ girls 4 boys or 4 girls 3 boys or 5 girls 2 boys
$={ }^{5} \mathrm{C}_{3} \times{ }^{7} \mathrm{C}_{4}+{ }^{5} \mathrm{C}_{4} \times{ }^{7} \mathrm{C}_{3}+{ }^{5} \mathrm{C}_{5} \times{ }^{7} \mathrm{C}_{2}$
$=10 \times 35+5 \times 35+1 \times 21$
$=350+175+21$
$=546$
(ii) Ways to select exactly three girls
$=3$ girls 4 boys
$={ }^{5} \mathrm{C}_{3} \times{ }^{7} \mathrm{C}_{4}=350$
(iii)Ways of arranging 3 girls and 4 boys if all girls and boys stand together
$=2!\times 3!\times 4!$
$=2 \times 6 \times 24$
$=288$
Total ways of selecting and arranging
$=288 \times 350$
$=100800$
(iv)Ways to arrange boys $=4$ !
B _ B _ B _ B
Ways to arrange girls $=3$ !
Total ways of selecting and arranging
$=4!\times 3!\times 350$
$=24 \times 6 \times 350$
$=50400$
