Answer to Question #176341 in Discrete Mathematics for Preethi

Question #176341

How many bit strings of length 14 contains the following types

(I) atmost three is?

(ii) an equal number of 0s and 1s

(iii) an odd number of 1s



1
Expert's answer
2021-03-30T10:54:01-0400

I. Atmost three

C(n,r)=n!(nr)!r!n=14r3=C(14,0)+C(14,1)+C(14,2)+C(14,3)=1+14+91+364=470C(n,r)=\frac{n!}{(n-r)!r!}\\ n=14\\ r≤3\\ =C(14,0)+C(14,1)+C(14,2)+C(14,3)\\ =1+14+91+364=470

II. An equal number of 0s and 1s. This means that there will be 7 1s and 7 0s.

n=14r=7=C(14,7)=3432n=14\\ r=7\\ =C(14,7)\\ =3432

III. An odd number of 1s. This means that 1s can be 1,3,5,7,9,11,13.


=C(14,1)+C(14,3)+C(14,5)+C(14,7)+C(14,9)+C(14,11)+C(14,13)=14+364+2002+3432+2002+364+14=8192=C(14,1)+C(14,3)+C(14,5)+C(14,7)+C(14,9)+C(14,11)+C(14,13)\\ =14+364+2002+3432+2002+364+14\\ =8192


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