Sum
Nine friends decide to go for a picnic in two groups. One group decides to go by car and the other group decides to go by train. Find the number of different ways of doing so if there must be at least 3 friends in each group.
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Solution
Train  Car  Number of outcomes 

No. of friends 
3  6  `""^9"C"_3` 
4  5  `""^9"C"_4`  
5  4  `""^9"C"_5`  
6  3  `""^9"C"_6` 
Required number = `""^9"C"_3 + ""^9"C"_4 + ""^9"C"_5 + ""^9"C"_6`
= `(""^9"C"_4 + ""^9"C"_3) + (""^9"C"_6 + ""^9"C"_5) = ""^10"C"_4 + ""^10"C"_6`
= `(10!)/(6!4!) + (10!)/(4!6!)`
= 210 + 210
= 420
Concept: Properties of Combinations
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