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Binomial Coefficients and N Choose K.md

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+#Math #Probability
+
+# Problem
+
+Why does n choose k, or $\frac{n!}{k!(n-k)!}$ generate the coefficient for $x^ky^{n-k}$ in $(x+y)^n$?
+
+# Explanation
+
+Let’s see what happens when expanding $(x+y)^4$:
+
+$$
+(x+y)^4\\
+=(x+y)(x+y)(x+y)(x+y)\\
+=xxxx+\\
+yxxx+xyxx+xxyx+xxxy+\\
+yyxx+yxyx+yxxy+xyyx+xyxy+xxyy+\\
+xyyy+yxyy+yyxy+yyyx+\\
+yyyy
+$$
+
+When expanding, notice the number of terms with k of x (and likewise 4-k of y) is the number of combinations of 4 choose k, as you choose k slots to put k x’s in out of 4 slots. Therefore, $(x+y)^n={n \choose 0}x^0y^n+{n \choose 1}x^1y^{n-1}...+{n \choose n-1}x^{n-1}y^1+{n \choose n}x^ny^0$