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Question

Let f(n) denotes the number of different ways in which the positive integer n can be expressed as the sum of 1s and 2s. For example f(4)=5, since
4=2+2=2+1+1=1+2+1=1+1+2=1+1+1+1.

Then which of the following(s) is (are) CORRECT ?

A
f(6)=13
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B
f(f(6))=377
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C
f(f(6))=370
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D
f(6)=11
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Solution

The correct options are
A f(6)=13
B f(f(6))=377
6=0(2)+6(1)=1(2)+4(1)=2(2)+2(1)=3(2)+0(1)

Total number of permutations
=6!6!+5!4!+4!2! 2!+3!3!=13f(6)=13

f(f(6))=f(13)13=0(2)+13(1)=1(2)+11(1)=2(2)+9(1)=3(2)+7(1)=4(2)+5(1)=5(2)+3(1)=6(2)+1(1)

Total number of permutations
=13!13!+12!11!+11!2! 9!+10!3! 7!+9!4! 5!+8!5! 3!+7!6!

=1+12+55+120+126+56+7=337f(f(6))=377


Alternate solution :
We know that,
f(1)=1f(2)=23=1+1+1;1+2;2+1f(3)=3f(4)=55=5(1);2(2)+1(1);1(2)+3(1)f(5)=1+3!2! 1!+4!3! 1!=8

From the above, we can see a pattern
f(5)=f(4)+f(3)f(n)=f(n1)+f(n2)
Therefore, f(6)=f(5)+f(4)=8+5=13
f(f(6))=f(13)

Now,
f(13)=f(12)+f(11)f(13)=2f(11)+f(10)f(13)=3f(10)+2f(9)f(13)=5f(9)+3f(8)f(13)=8f(8)+5f(7)f(13)=13f(7)+8f(6)f(13)=21f(6)+13f(5)f(13)=21×13+13×8f(13)=13×29=377


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