Question-49036

Question Number 49036 by Tinkutara last updated on 01/Dec/18

Commented by prakash jain last updated on 02/Dec/18

2^(n−1) gave at least 1 wrong answer (include student with 2 wrong answers) 2^(n−2) gave at least 2 wrong amswers (only one additional wrong answer should be counted) ... 2^(n−k) gave k wrong answer S=2^(n−1) +2^(n−2) +...+2^(n−k) 2^(n−1) (((1−((1/2))^k )/(1−(1/2))))=2^n ×((2^k −1)/2^k ) example n=6, k=3 32 − 1 or more wrong 16 − 2 of more wrong 8 − 3 or more wrong total wrong answer 32+16+8=56

$$\mathrm{2}^{{n}−\mathrm{1}} \:{gave}\:{at}\:{least}\:\mathrm{1}\:{wrong}\:{answer} \\ $$$$\left({include}\:{student}\:{with}\:\mathrm{2}\:{wrong}\:{answers}\right) \\ $$$$\mathrm{2}^{{n}−\mathrm{2}} \:{gave}\:{at}\:{least}\:\mathrm{2}\:{wrong}\:{amswers}\: \\ $$$$\left({only}\:{one}\:{additional}\:{wrong}\:{answer}\right. \\ $$$$\left.{should}\:{be}\:{counted}\right) \\ $$$$… \\ $$$$\mathrm{2}^{{n}−{k}} \:{gave}\:{k}\:{wrong}\:{answer} \\ $$$${S}=\mathrm{2}^{{n}−\mathrm{1}} +\mathrm{2}^{{n}−\mathrm{2}} +…+\mathrm{2}^{{n}−{k}} \\ $$$$\mathrm{2}^{{n}−\mathrm{1}} \left(\frac{\mathrm{1}−\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{{k}} }{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}}\right)=\mathrm{2}^{{n}} ×\frac{\mathrm{2}^{{k}} −\mathrm{1}}{\mathrm{2}^{{k}} } \\ $$$${example} \\ $$$${n}=\mathrm{6},\:{k}=\mathrm{3} \\ $$$$\mathrm{32}\:−\:\mathrm{1}\:{or}\:{more}\:{wrong} \\ $$$$\mathrm{16}\:−\:\mathrm{2}\:{of}\:{more}\:{wrong}\: \\ $$$$\mathrm{8}\:\:\:\:−\:\mathrm{3}\:{or}\:{more}\:{wrong} \\ $$$${total}\:{wrong}\:{answer}\:\mathrm{32}+\mathrm{16}+\mathrm{8}=\mathrm{56} \\ $$$$ \\ $$

Commented by prakash jain last updated on 02/Dec/18

$$\mathrm{Book}\:\mathrm{answer}\:\mathrm{will}\:\mathrm{be}\:\mathrm{correct}\:\mathrm{if}\:{n}={k}. \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 03/Dec/18

$${thank}\:{you}\:{sir}… \\ $$

Commented by Tinkutara last updated on 03/Dec/18

Thank you very much Sir! I got the answer. ��

Answered by tanmay.chaudhury50@gmail.com last updated on 02/Dec/18

put i=1 2^(n−1) students give 1wrong answer 2^(n−2) students give 2wrong ansrwer 2^(n−3) students give 3 wrong answer ... ... 2^(n−k) students give k wrong answer total wrong answer S=2^(n−1) ×1+2^(n−2) ×2+2^(n−3) ×3...+2^(n−k) ×k S×2^(−1) = 2^(n−2) ×1 +2^(n−3) ×2.. +2^(n−k) ×(k−1)+2^(n−k−1) ×k substruct (S−(S/2))=(2^(n−1) +2^(n−1) +...+2^(n−1) n times)−2^(n−k−1) ×k (S/2)=n2^(n−1) −2^(n−k−1) ×k S=n2^n −2^(n−k) ×k

$${put}\:{i}=\mathrm{1} \\ $$$$\mathrm{2}^{{n}−\mathrm{1}} {students}\:{give}\:\mathrm{1}{wrong}\:{answer} \\ $$$$\mathrm{2}^{{n}−\mathrm{2}} \:{students}\:\:{give}\:\mathrm{2}{wrong}\:{ansrwer} \\ $$$$\mathrm{2}^{{n}−\mathrm{3}} \:{students}\:\:{give}\:\mathrm{3}\:{wrong}\:{answer} \\ $$$$… \\ $$$$… \\ $$$$\mathrm{2}^{{n}−{k}} \:{students}\:{give}\:{k}\:{wrong}\:{answer} \\ $$$${total}\:{wrong}\:{answer} \\ $$$$\:\:\:\:\:\:{S}=\mathrm{2}^{{n}−\mathrm{1}} ×\mathrm{1}+\mathrm{2}^{{n}−\mathrm{2}} ×\mathrm{2}+\mathrm{2}^{{n}−\mathrm{3}} ×\mathrm{3}…+\mathrm{2}^{{n}−{k}} ×{k} \\ $$$${S}×\mathrm{2}^{−\mathrm{1}} =\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}^{{n}−\mathrm{2}} ×\mathrm{1}\:\:\:\:+\mathrm{2}^{{n}−\mathrm{3}} ×\mathrm{2}..\:\:+\mathrm{2}^{{n}−{k}} ×\left({k}−\mathrm{1}\right)+\mathrm{2}^{{n}−{k}−\mathrm{1}} ×{k} \\ $$$${substruct} \\ $$$$\left({S}−\frac{{S}}{\mathrm{2}}\right)=\left(\mathrm{2}^{{n}−\mathrm{1}} +\mathrm{2}^{{n}−\mathrm{1}} +…+\mathrm{2}^{{n}−\mathrm{1}} \:{n}\:{times}\right)−\mathrm{2}^{{n}−{k}−\mathrm{1}} ×{k} \\ $$$$\frac{{S}}{\mathrm{2}}={n}\mathrm{2}^{{n}−\mathrm{1}} −\mathrm{2}^{{n}−{k}−\mathrm{1}} ×{k} \\ $$$${S}={n}\mathrm{2}^{{n}} −\mathrm{2}^{{n}−{k}} ×{k} \\ $$

Commented by Tinkutara last updated on 02/Dec/18

But answer is independent of k.

Commented by tanmay.chaudhury50@gmail.com last updated on 02/Dec/18

$$\mathrm{0}{k}\:{let}\:{others}\:{try}\:{or}\:{you}\:{post}\:{answer}\:{in}\:{details} \\ $$

Commented by Tinkutara last updated on 02/Dec/18

Ans is 2^n −1 I don′t understand the solution given

$${Ans}\:{is}\:\mathrm{2}^{{n}} −\mathrm{1} \\ $$$${I}\:{don}'{t}\:{understand}\:{the}\:{solution}\:{given} \\ $$

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