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# Interesting Combinatorics Problem! :D

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Source: Mu Alpha Theta

Check their website over here: https://mualphatheta.org/

Evaluate:

$$\binom{15}{0} + 3\binom{15}{1}+....+ 31\binom{15}{15}$$

Your answer should be in the form of 2^n.

Hint:https://web2.0calc.com/questions/double-counting

Jul 9, 2022

### 3+0 Answers

#1
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listfor(n, 0, 15, ((2*n + 1)*(15 nCr n))==(1, 45, 525, 3185, 12285, 33033, 65065, 96525, 109395, 95095, 63063, 31395, 11375, 2835, 435, 31)>>>Sum==524,288

Jul 9, 2022
#2
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Now solve this https://web2.0calc.com/questions/need-help-with-this_18

Jul 9, 2022
#3
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Interesting Problem!!!!!!

$$\text{ Let } S=$$$$15C0+3(15C1)+5(15C2)+7(15C3)+...+31(15C15)$$

Then,  $$S=(15C15)+3(15C14)+...+31(15C0)$$     (Using the property that: $$nCk = nC(n-k)$$ ).

Adding the expressions above, to get:

$$2S=(15C0)(31+1)+(15C1)(3+29)+...+(15C15)(31+1)$$

Hence, $$2S=32(15C0+15C1+15C2+...+15C15)=32*2^{15}=2^5*2^{15}=2^{20}$$

Therefore, $$S=2^{19}$$

Jul 10, 2022