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3 August, 23:50

Recall that a 6-bit string is a bit strings of length 6, and a bit string of weight 3, say, is one with exactly three 1's. How many 6-bit strings are there? How many 6-bit strings have weight 0? How many 6-bit strings have weight 1? How many 6-bit strings have weight 3? How many 6-bit strings have weight 5? How many 6-bit strings have weight 6? How many 6-bit strings have weight 7?

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  1. 4 August, 01:11
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    1 ... Total number of 6 bit strings is 64

    2. Number of 6-bit strings with weight of 0 is 1

    3. Number of 6-bit strings with weight of 1 is 6

    4. Number of 6-bit strings with weight of 3 is 20

    5. Number of 6-bit strings with weight of 5 is 6

    6. Number of 6-bit strings with weight of 6 is 1

    7. Number of 6-bit strings with weight of 7 is 0

    Step-by-step explanation:

    A bit string is a string that contains 0 and 1 only

    1. Total number of 6 bit strings is 2^6 = 64

    2. Number of 6 bit strings with weight 0 is 1

    Explanation

    Weight 0 means a string with no occurrence of 1

    Here, we are only interested in occurrence and not order of occurrence

    We apply combination formula for this

    nCr = n! / (n-r) ! r!

    n = 6 and r = 0 i. e. no occurrence of 1

    6C0 = 6! / (6-0) !0!

    6C0 = 6!/6!0!

    6C0 = 1

    Hence, the number of string with weight 0 (i. e. no occurrence of 1) is 1

    3. Number of string with weight 1 is 6

    Explanation

    Weight 0 means a string with exactly 1 occurrence of '1'

    Here, we are only interested in occurrence and not order of occurrence

    We apply combination formula for this

    nCr = n! / (n-r) ! r!

    n = 6 and r = 1

    6C1 = 6! / (6-1) !1!

    6C1 = 6!/5!1!

    6C1 = 6

    Hence, the number of string with weight 6

    4. Number of string with weight 3 is 20

    Explanation

    n = 6 and r = 3

    6C3 = 6! / (6-3) !3!

    6C3 = 6!/3!3!

    6C3 = 20

    Hence, the number of string with weight 3 is 20

    5. Number of string with weight 5 is 6

    Explanation

    n = 6 and r = 5

    6C5 = 6! / (6-5) !5!

    6C5 = 6!/1!5!

    6C5 = 6

    Hence, the number of string with weight 5 is 6

    6. Number of string with weight 6 is 1

    Explanation

    n = 6 and r = 6

    6C6 = 6! / (6-6) !6!

    6C6 = 6!/0!6!

    6C6 = 1

    Hence, the number of string with weight 6 is 1

    7. Number of string with weight 7 is 0

    Weight of 7 means that a string that has 7 occurrence of 1

    The total length of a 6 bit is 6

    Since 6 is less than 7, there's no way a bit of weight 7 can occur.

    So, the right answer for this is 0.
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