(Q). There are infinite number of letters F, L and Y. In how many ways we can choose ‘n’ letters such that the word ‘FLY’ can not be formed?
Solution:-
The word ‘FLY’ can not be formed if the ‘n’ selected letters do not contain at least one of F, L and Y.
when ‘n’ letters are selected from F or L
every box has 2 ways of filling (either F or L)
Hence Number of ways of selecting ‘n’ letters which are F or L = 2ⁿ including the case when all the ‘n’ letters are F or all are L ……………………….. case(i)
Similarly, Number of ways of selecting ‘n’ letter which are F or Y = 2ⁿ including the case when all the ‘n’ letters are F or all are Y………………………. case(ii)
& number of ways of selecting ‘n’ letters which are L or Y = 2ⁿ including the case when all the ‘n’ letters are L or all are Y……………………………..case(iii)
Similarly, number of ways of selecting ‘n’ letters when all are F, L and Y are 1ⁿ = 1 in each case
Thus required number of ways
= 2ⁿ + 2ⁿ + 2ⁿ – 1 – 1 – 1
= 3(2ⁿ – 1) Answer