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Question

If 4 numbers are to be selected from 1,2,3,......25 such that they are be in AP, then number of ways of selecting numbers are

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Solution

an=a+(n1)d
25=1+(41)d
d=8
maximum difference between the numbers can be 8.
If we take maximum difference (8) then possible set of numbers are (1,9,17,25)
taking d=7 we get
(1,8,15,22),(2,9,16,23),(3,10,17,24),(4,11,18,25)
similarly taking d=6 we get
(1,7,13,19),(2,8,14,20),(3,9,15,21),.........(7,13,19,25)
every time reducing value of d with 1 we get 3 more set of numbers, so with d=1 we get 22 set of numbers
So total number of ways of selecting numbers are
22+19+16+......+1=92

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