There are 10 spaniels, 14 retrievers and 6 poodles at a dog show. 7 dogs are selected to go through to
the final.
(i) How many selections of 7 different dogs can be made if there must be aat least 1 spaniel, at least
2 retrievers and at least 3 poodles?
[4]
2 spaniels, 2 retrievers and 3 poodles go through to the final. They are placed in a line.
(ii) How many different arrangement of these 7 dogs are there if the spaniels stand together and the
retrievers stand together?
w
[3]
(iii) How many different arrangements of these 7 dogs are there if no poodle is next to another poodle?
[3]
Answers
Given : There are 10 spaniels, 14 retrievers and 6 poodles at a dog show
7 dogs are selected to go through to the final.
To Find : How many selections of 7 different dogs can be made if there must be at least 1 spaniel, at least 2 retrievers and at least 3 poodles
Solution:
at least 1 spaniel, at least 2 retrievers and at least 3 poodles
Three possible cases :
2 spaniel, 2 retrievers 3 poodles
1 spaniel, 3 retrievers 3 poodles
1 spaniel, 2 retrievers 4 poodles
2 spaniel, 2 retrievers 3 poodles ways
¹⁰C₂.¹⁴C₂.⁶C₃ = 81,900
1 spaniel, 3 retrievers 3 poodles
¹⁰C₁.¹⁴C₃.⁶C₃ = 72,800
1 spaniel, 2 retrievers 4 poodles
¹⁰C₁.¹⁴C₂.⁶C₄ = 13,650
Total = 81,900 + 72,800 + 13,650 = 1,68,350
Similarly others can be done
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