Permutations an...

$$12$$ people came for a carroms tournament. All were divided into pairs of $$2$$ each. After all the matches if it found that half of the selected pairs had to play three matches to decide the winner in a best of three process, how many matches have been held?
 

If $$^{19}C_{3r} = ^{19}C_{r+3} $$, then r is equal to:

How many alphabets need to be there in a language if one were to make $$1$$ million distinct $$3$$ digit initials using the alphabets of the language?

 After every get-together every person present shakes the hand of every other person. If there were 105 handshakes in all, how many persons were present in the party?

$$8064$$ is resolved into all possible product of two factors. Find the number of ways in which this can be done?

How many straight lines can be formed from $$11$$ points in a plane out of which no three points are collinear?

A college offers $$7$$ courses in the morning and $$5$$ in the evening. Find possible number of choices with the student who want to study one course in the morning and one in the evening.

When six fair coins are tossed simultaneously, in how many of the outcomes will at most three of the coins turn up as heads?

The number of ways in which four letters can be selected from the word 'DEGREE' is

If $$^{15}C_{3r} = ^{15}C_{r+3}$$, then find the value of $$r$$: