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Find the number of ways in which 3 distinct numbers can be selected from the set `{3^(1),3^(2),3^(3),..,3^(100),3^(101)}` so that they form a G.P. |
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Answer» Correct Answer - 2500 Let three numbers selected be `3^(a),3^(b),3^(c )` which are in G.P. `therefore (3^(b))^(2)=(3^(a))(3^(c ))` `implies 2b=a+c` `implies a,b,c ` are in A.P. Thus, selecting three number in G.P. from given set is equivalent to selecting 3 numbers from {1,2,3,..,101} which are in A.P. Now, a,b,c are in A.P. if either a and c are odd or a and c are even. Number of ways of selecting two odd numbers is `. ^(51)C_(2)` and those of selecting two even numbers is `.^(50)C_(2)`. Onece a and c are selected, b is fixed. Hence total number of ways `= .^(51)C_(2)+ .^(50)C_(2)=1275+1225=2500` |
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