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Sagot :
a) Number of bit string = 220
b) Number of bit string = 299
c) Number of bit string = 4017
d) Number of bit string = 924
What is Combination?
Combinations are mathematical operations that count the number of potential configurations for a set of elements when the order of the selection is irrelevant.
Given:
a) n= 12, r= 3
[tex]^{12}C_3[/tex] = 12! / 3! 9 !
= 12 x 11 x 10 x 9! / 3 x 2 x 9!
= 220
b) n= 12, r ≤ 3
[tex]^{12}C_3[/tex] = 12! / 3! 9 !
= 12 x 11 x 10 x 9! / 3 x 2 x 9!
= 220
[tex]^{12}C_2[/tex] = 12! / 2! 10 !
= 12 x 11 x 10! / 2 x 10!
= 66
[tex]^{12}C_1[/tex] = 12! / 1! 11!
= 12
[tex]^{12}C_0[/tex] = 12! / 0! 12 !
= 12 x 11 x 10 x 9! / 3 x 2 x 9!
= 1
So, total number of bit strings = 220 +66 + 12 +1 =299
c) n= 12, r ≥ 3
[tex]^{12}C_3[/tex] = 12! / 3! 9 !
= 12 x 11 x 10 x 9! / 3 x 2 x 9!
= 220
[tex]^{12}C_4[/tex] = 495
[tex]^{12}C_5[/tex] = 792
[tex]^{12}C_6[/tex] = 924
[tex]^{12}C_7[/tex] = 792
[tex]^{12}C_8[/tex] = 495
[tex]^{12}C_9[/tex] = 220
[tex]^{12}C_{10[/tex] = 66
[tex]^{12}C_{12[/tex] = 12
[tex]^{12}C_{12[/tex] = 1
So, total String= 4017
d) n= 12 and r= 6
[tex]^{12}C_6[/tex] = 12!/ 6! 6!
= 12 x 11 x 10 x 9 x 8 x 7 x 6! / 6! 6!
= 924
Learn more about Combination here:
https://brainly.com/question/28720645
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