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Question
Compute 97
Solution
97 = (10 – 1)7
= 7C0 (10)7 (– 1)0 + 7C1 (10)7–1 (– 1)1 + 7C2 (10)7–2 (– 1)2 + 7C3 (10)7–3 (– 1)3 + 7C4 (10)7–4 (– 1)4 + 7C5 (10)7–5 (– 1)5 + 7C6 (10)7–6 (– 1)6 + 7C7 (10)7–7 (– 1)7
= `1 xx 10^7 xx 1 + 7 xx 10^6 xx - 1 + (7 xx 6)/(1 xx 2) xx 10^5 xx 1 + (7 xx 6 xx 5)/(1 xx 2 xx 3) xx 10^4 xx - 1 + ""^7"C"_3 xx 10^3 xx 1 + ""^7"C"_2 xx 10^2 xx -1 + ""^7"C"_1 xx 10^1 xx 1 + ""^7"C"_7 xx - 1`
= `10^7 - 7 xx 10^6 + 21 xx 10^5 - 35 xx 10^4 + (7 xx 6 xx 5)/(1 xx 2 xx 3) xx 10^3 - (7 xx 6)/(1 xx 2) xx 10^2 + 7 xx 10 - 1`
= 103(104 – 7 × 103 + 21 × 102 – 35 × 10 + 35) – 21 × 100 + 70 – 1
= 103(10000 – 7000 + 2100 – 350 + 35 ) – 2100 + 70 – 1
= 103(12135 – 7350) – 2031
= 103 × 4785 – 2031
= 4785000 – 2031
= 4782969
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