Quod Erat Demonstrandum

2015/01/23

某數算題

Filed under: mathematics,NSS — johnmayhk @ 5:34 下午
Tags: , ,

Just reply to a F.5C student on a basic core mathematics question (on P.5.38):

There are 8 outstanding students from junior forms and 9 outstanding students from senior forms in a school this year. 5 out of these 17 students are now selected for an overseas exchange programme. Find the number of combinations of selecting at least 1 student from junior forms and 1 from senior forms.

Here is the ‘so-called’ solution from a student:

_8C_1\times _9C_1\times _{15}C_3

as the student claimed, select 1 from junior, _8C_1 ways; select 1 from senior, _9C_1 ways; then select the remaining 3 students from the remaining 15 students, _{15}C_3 ways, hence, the total number of combination should be _8C_1\times _9C_1\times _{15}C_3, right?

Sorry, it is incorrect. (more…)

廣告

2015/01/12

just a core math question involving variance

Filed under: mathematics,NSS — johnmayhk @ 10:35 下午

Just reply to a student about a core math question used in school exam.

Question

Let m, v be the mean and variance of {2x_1, 2x_2, 2x_3} respectively.

Show that the mean and variance of the set

{x_1+x_1,x_1+x_2,x_1+x_3,x_2+x_1,x_2+x_2,x_2+x_3,x_3+x_1,x_3+x_2,x_3+x_3}

are m and \frac{v}{2} respectively.

Proof

(Mean) For the original set, (more…)

2015/01/09

推特老題

Filed under: HKALE,Pure Mathematics — johnmayhk @ 3:26 下午
Tags: ,

早前在推特見:

johnmayhk-question-in-twitter

他們討論著,並表示不知道如何得出如此美妙式子。

如何美妙?修 M1,M2 的同學應該不難察覺,式子包含二項系數(binomial coefficients),

11
121
1331
14641

並且,把等號左邊的項加起來,得右邊單項式作分子云云。

但有修純數的人,相信很快察覺,那不過是一道有關部份分式(partial fractions)的基本題目(搖頭中)。Okay, (more…)

2015/01/08

被 8 整除

Filed under: Fun,mathematics — johnmayhk @ 2:08 下午
Tags: , ,

1.問題

證明對於任何正整數 n

n^4+2n^3+3n^2+2n

必可被 8 整除。

2.解答

(a)

本來利用 Mathematical Induction (M.I.) 來證明是很輕易的,但現在的 M2 課程已刪除了整除性,相信中四五的同學,要解題不易。

只寫 P(k)\Rightarrow P(k+1) 這步: (more…)

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