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Calculate 11C11+12C11+13C11+---+98C11+99C11

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zoeeyz | Student, College Freshman | eNoter

Posted July 5, 2013 at 5:25 PM via web

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Calculate 11C11+12C11+13C11+---+98C11+99C11

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted July 5, 2013 at 6:00 PM (Answer #1)

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You need to use Pascal's formula, such that:

`C_n^k = C_(n-1)^k + C_(n-1)^(k-1)`

Reasoning by analogy yields:

`C_99^11 + C_99^12 = C_100^12`

`C_98^11 + C_98^12 = C_99^12`

..............................................

`C_12^11 + C_12^12 = C_13^12`

`C_11^11 = C_12^11`

Adding the members both sides, yields:

`C_99^11 + C_98^11 + ... + C_12^11 + C_11^11 + C_99^12 + C_98^12 + ... + C_12^12 = C_100^12 + C_99^12 + ... + C_13^12 + C_12^11`

Reducing duplicate members both sides, yields:

`C_99^11 + C_98^11 + ... + C_12^11 + C_11^11 = C_100^12`

Hence, evaluating the given summation, using Pascal's formula, yields` C_99^11 + C_98^11 + ... + C_12^11 + C_11^11 = C_100^12.`

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