find the 20th term in the expansion of (a+b)^22
That would be
22C19 a^3 b^19 = 22C3 a^3 b^19 = 1540a^3b^19
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To find the 20th term in the expansion of `(a+b)^22`, we can use the binomial theorem. According to the binomial theorem, the terms in the expansion will follow the pattern:
(22 choose 0) * a^22 * b^0 + (22 choose 1) * a^21 * b^1 + (22 choose 2) * a^20 * b^2 + ... + (22 choose n) * a^(22-n) * b^n + ...
The term number (in this case, 20) corresponds to the term with `(22 choose n) * a^(22-n) * b^n`. We need to determine the value of `n`.
To do this, we can use the formula for the rth term of a binomial expansion: (n choose r-1). Here, r is the term number (20) and n is the power of the binomial (22). So, using this formula:
(n choose r-1) = (22 choose 20-1) = (22 choose 19) = 22! / (19! * 3!)
To calculate this value, we first need to determine the factorial of 22, 19, and 3:
22! = 22 * 21 * 20 * 19!
Now we can substitute the values into the formula:
(22 choose 19) = (22 * 21 * 20 * 19!) / (19! * 3!)
Simplifying the expression:
(22 * 21 * 20) / 3 = 1540
Therefore, the 20th term in the expansion of `(a+b)^22` is `1540 * a^3 * b^19`.