GMAT一个质因子的,一个概率的第一题Foreverypositiveevenintegern,thefunctionh(n)isdefinedtobetheproductofalltheevenintegersfrom2ton,inclusive.Ifpisthesmallestprimefactorofh(100)+1,thenpisA.between2and1

Question

GMAT 一个质因子的,一个概率的
第一题
For every positive even integer n,the function h(n) is defined to be the product of all the even integers from 2 to n,inclusive.If p is the smallest prime factor of h(100) +1,then p is
A.between 2 and 10
B.between 10 and 20
C.between 20 and 30
D.between 30 and 40
E.greater than 40
第二题
Tanya prepared 4 different letters to be sent to 4 different addresses.For each letter,she prepared an envelope with its correct address.If the 4 letters are to be put into the 4 envelopes at random,what is the probability that only 1 letter will be put into the envelope with its correct address.

Answer 1

第一题,选E, 因为50以下的素数的2倍肯定出线在100!(双阶乘知道吧?就是h(100)）,于是被这样的素数除,肯定余1,所以50以下的素数不可能是因子.
第二题,1/3
记Ai={第i封信放对了}, Ai^c为对立事件, AB表示交, AUB表示并
则所求为4P(A1(A2^c)(A3^c)(A4^c))
=4P(A1)P((A2^c)(A3^c)(A4^c) | A1)
=4P(A1)P((A2^c)(A3^c)(A4^c))
第二个概率就是n=3时的放错信封问题的经典概率,可以参考任何一本概率书,查看.
最简单可以用概率加法公式,也就是容斥原理来处理.（=1/2!-1/3!+1/4!-...+(-1)^(n-1)1/n!）
最后得=4*1/4*(1/2-1/6)=1/3