Answer to Question #89931 in Discrete Mathematics for curtis

Question

How many 20 digit binary numbers have four 1's in them?

Answer 1

The number of 20 digits binary numbers with exactly k 1's in them is C(19,k-1) (where C(n,m) is binomial coefficient), since first digit of number is 1 and other k-1 one's are placed in 19 remaining positions.

So there are 1 number with exactly one digit 1;

C(19,2-1) = C(19,1) = 19 numbers with exactly two digits 1;

C(19,3-1) = C(19,2) = 19*18/(1*2) = 171 numbers with exactly three digits one;

and C(19,4-1) = C(19,3) = 19*18*17/(1*2*3) = 969 numbers with exactly four digits one.

The number of 20 digit binary numbers is 2¹⁹, since first digit is always 1.

The number of 20 digit binary numbers with not fewer than four 1's is 2¹⁹ -1 -19-171 = 524097.

The number of 20 digit binary numbers with exactly 4 1's is 969.

Answer: There are 969 twenty digit binary numbers with exactly four 1's in them, and 524097 with at least four 1's in them.

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In case first digit of binary number can be 0, solution is

The number of 20 digits binary numbers with exactly k 1's in them is C(20,k) (where C(n,m) is binomial coefficient), since k one's are placed in 20 positions.

So there are 1 number with exactly zero digit 1;

C(20,1) = 20 numbers with exactly one digit 1;

C(20,2) = 20*19/(1*2) = 190 numbers with exactly two digits 1;

C(20,3) = 20*19*18/(1*2*3) =1140 numbers with exactly three digits 1;

C(20,4) = 20*19*18*17/(1*2*3*4) = 4845 number with exactly four digits 1.

The number of 20 digit binary numbers is 2²⁰.

The number of 20 digit binary numbers with not fewer than four 1's is 2²⁰-1-20-190-1140 = 1047225.

Answer: There are 4845 twenty digit binary numbers with exactly four 1's in them, and 1047225 with at least four 1's in them.