Permutations and Combinations

Permutations and Combinations

Question

Insert A.M.’s between 7 and 71 in such a way that the 5th A.M. is 27. Find the number of A.M.’s.

Answer

Let   straight A subscript 1 comma space space straight A subscript 2 comma space............. comma space straight A subscript straight n be n A.M.'s inserted between 7 and 71.

∴  7, straight A subscript 1 comma space space straight A subscript 2 comma space...... comma space straight A subscript straight n comma space 71 are in A.P.
                                 a = 7,
Let d be the common difference
                    straight t subscript straight n plus 2 end subscript space equals space 71               rightwards double arrow      a + (n + 2 - 1)d = 71
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#6 {main}</pre>              ...(i)
Now,         straight A subscript 5 space equals space 27                        or              straight t subscript 6 space equals space 27
or            a + 5d = 27                      or           7 + 5d = 27
or                 5d = 20                       or                  d = 4                  ...(ii)
From (i) and (ii), we have
               fraction numerator 64 over denominator straight n plus 1 end fraction space equals space 4                    or                   n + 1 = 16
or                        n = 15

∴      15 A.M.'s are inserted between 7 and 11.                     

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