The value of [tex]^nC_0 + ^nC_3 + ^nC_6 + ^nC_9[/tex] is [tex]1 + \frac{n!}{6}[\frac{1}{(n - 3)!}+\frac{1}{120(n - 6)!} + \frac{1}{60480(n - 9)!}][/tex]
The expression is given as:
[tex]^nC_0 + ^nC_3 + ^nC_6 + ^nC_9[/tex]
The combination formula is:
[tex]^nC_r = \frac{n!}{(n - r)!r!}[/tex]
So, we have:
[tex]\frac{n!}{(n - 0)!0!} + \frac{n!}{(n - 3)!3!}+\frac{n!}{(n - 6)!6!} + \frac{n!}{(n - 9)!9!}[/tex]
Evaluate the factorials
[tex]\frac{n!}{(n - 0)!} + \frac{n!}{6(n - 3)!}+\frac{n!}{720(n - 6)!} + \frac{n!}{362880(n - 9)!}[/tex]
Evaluate the difference
[tex]\frac{n!}{n!} + \frac{n!}{6(n - 3)!}+\frac{n!}{720(n - 6)!} + \frac{n!}{362880(n - 9)!}[/tex]
Evaluate the quotient
[tex]1 + \frac{n!}{6(n - 3)!}+\frac{n!}{720(n - 6)!} + \frac{n!}{362880(n - 9)!}[/tex]
Factor out n!
[tex]1 + n![\frac{1}{6(n - 3)!}+\frac{1}{720(n - 6)!} + \frac{1}{362880(n - 9)!}][/tex]
Factor out 1/6
[tex]1 + \frac{n!}{6}[\frac{1}{(n - 3)!}+\frac{1}{120(n - 6)!} + \frac{1}{60480(n - 9)!}][/tex]
Hence, the value of [tex]^nC_0 + ^nC_3 + ^nC_6 + ^nC_9[/tex] is
[tex]1 + \frac{n!}{6}[\frac{1}{(n - 3)!}+\frac{1}{120(n - 6)!} + \frac{1}{60480(n - 9)!}][/tex]
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