渚嬶細8浜哄洿鐫槨愭鍚冮キ錛屽灝戠灝卞駭鏂瑰紡錛?/p>
Ans: P(8, 8)/8=7!
2錛?nbsp; 閲嶆帓鍒?br>a. 鏃犻檺閲嶆帓鍒楋細n涓笉鍚屽厓绱犱腑鍙杛涓寜嬈″簭鎺掑垪錛屾瘡涓厓绱犲彲鍙栨棤闄愭錛屾繪暟涓簄r銆?/p>
b. 鏈夐檺閲嶆帓鍒楋細r涓笉鍚岃壊褰╃悆鏀懼叆n涓爣鍙風殑鐩掑瓙錛岀i縐嶅僵鐞冩湁ri涓紝鎬繪暟涓篜(n, r) / (r1!* r2!*… rt!)
3錛?nbsp; 闈為噸緇勫悎錛氭瘡涓厓绱犳渶澶氬嚭鐜頒竴嬈?br>C(n, r)
4錛?nbsp; 閲嶇粍鍚?br>N涓笉鍚岀殑鍏冪礌涓彇r涓厓绱狅紝鍏佽閲嶅鍙栵紝涓嶈冭檻欏哄簭銆傛繪暟涓篊(n+r-1, r)
5錛?nbsp; 姣嶅嚱鏁?br>a. 寮曞嚭錛?/p>
(x1+ x2+… +xk)n鐨勭粍鍚堟暟瀛︽剰涔夋槸灝唍涓棤鍖哄埆鐨勭悆鏀懼叆k涓紪鍙蜂笉鍚岀殑鐩掑瓙閲岋紝姣忎釜鐩掑瓙鐞冩暟涓嶉檺銆傚欏瑰紡灞曞紑鍚庯紝x1n1 *x2n2*…* xknk閫氳繃騫傚彲浠ヨ〃紺轟竴緇勮В銆傝岃繖涓」鐨勭郴鏁頒負
C(n, n1)* C(n-n1, n2)*…* C(n-n1-n2-…-nk-1, nk)=n!/ (n1!n2!…nk! )
鍚勭郴鏁頒箣鍜屼負kn銆?/p>
b. 鏅氭瘝鍑芥暟
涓涓簭鍒梴an}錛岀Оa0+a1x+ a2x2+…+ anxn+…榪欎釜澶氶」寮忎負{an}鐨勬櫘閫氭瘝鍑芥暟銆?/p>
渚?錛氾紙澶╁鉤縐扮墿闂錛夋湁璐ㄩ噺n1錛宯2…nk鏁存暟鍏嬬殑鐮濈爜錛岃縐癷鍏嬬墿浣擄紝鐗╀綋鍦ㄥ乏錛岀牆鐮佸湪鍙熾傚叡鏈夊灝戠涓嶅悓鐨勭О娉曪紵
瑙ｏ細璁炬湁ai縐嶆柟娉曪紝鍒?/p>
(1+xn1) (1+xn2)…(1+xnk)=鈭?ai xi
1琛ㄧず(1+xnj)涓彁渚?錛岀牆鐮乶j娌℃湁鐢ㄤ笂銆?/p>
ai涓烘墍姹傘?/p>
娉細澶氶」寮忓睍寮鍚庯紝榪樺彲浠ョ湅鍑鴻兘縐板嚭鍝簺閲嶉噺
緇忛獙錛氬緇堣寰楋紝涓涓嫭鍙峰唴涓嬈′粎鏈変竴涓」琚彇錛岀敤浜庢彁渚涚粰灞曞紑寮忕殑鏌愪竴欏?/p>
渚?錛氾紙閲嶅鍙栫墿錛夋湁n縐嶄笉鍚岀殑鐗╁搧錛屾瘡縐嶇墿鍝佸垎鍒兘鏈澶氬彇b1錛宐2… bn浠躲備粠涓彲閲嶅鐨勫彇r浠剁墿鍝佹湁澶氬皯縐嶄笉鍚岀殑鍙栨硶錛?/p>
瑙ｏ細璁炬湁ar縐嶄笉鍚岀殑鍙栨硶銆傚垯
(1+x+x2+…+xb1) (1+x+x2+…+xb2)… (1+x+x2+…+xbn)=a0+a1x+ a2x2+…+ arxr+…
灞曞紑寮忎腑xr鐨勬潵婧恱m1xm2…xmn=xr
浜庢槸鎴愪簡閲嶇粍鍚堥棶棰橈紝絳旀涓篊(n+r-1, r)
渚?錛氭暣鏁版媶鍒嗭細鏁存暟r鎷嗗垎鎴恔1錛宬2… km鐨勫拰錛宬i鍏佽鏈澶氶噸澶峮i嬈°傛眰鎷嗗垎鏂規鏁般?/p>
瑙ｏ細榪欐槸姹俴1b1+ k2b2+…+ kmbm=r鐨勪笉瀹氭柟紼嬬殑闈炶礋鏁存暟瑙ｇ殑涓暟錛?<= bi<= ni 銆?/p>
鑰冭檻(1+xk1+xk1*2+xk1*3+…+xk1*n1)( 1+xk2+xk2*2+xk2*3+…+xk2*n2)…( 1+xkn+xkn*2+xkn*3+…++xkm*nm)
鍒欑瓟妗堟槸xr鐨勭郴鏁?/p>
c. 鎸囨暟姣嶅嚱鏁?/p>
N涓厓绱犱腑錛宎i閲嶅浜唍i嬈★紝姹備粠涓彇r涓厓绱犵殑鎺掑垪鏁頒負br銆?/p>
璁懼彇mi涓猘i錛屸垜mi=r銆傚垯鐩鎬簰涓嶅悓鐨勬帓鍒楁暟涓簉! / ∏mi!
鍒欏浜庢墍鏈夌殑mi鐨勬媶鍒嗘柟娉曪紝br=鈭? r! / ∏mi! )
渚?錛氳嫢鏈?涓厓绱狅紝鍏朵腑a1閲嶅浜?嬈★紝a2閲嶅浜?嬈★紝a3閲嶅浜?嬈°傛眰浠庝腑鍙栧嚭4涓厓绱犵殑鎺掑垪鏁般?/p>
瑙ｏ細鍏堟瀯閫犳櫘閫氭瘝鍑芥暟
G(x)=(1+x+ x2+x3) (1+x+ x2) (1+x+ x2+x3)
X4鐨勭郴鏁頒負10錛岃鏄庡彇4涓厓绱犵殑緇勫悎鏁頒負10銆傝繖鐩稿綋浜庝笂闈㈡墍璇寸殑瀵逛簬mi鐨勬媶鍒嗘柟娉曘?br>4 = 1+0+3 = 0+1+3 = 2+0+2 = 1+1+2 = 0+2+2 = 3+0+1 = 2+1+1 = 1+2+1 = 3+1+0 = 2+2+0
浠ｅ叆br=鈭? r! / ∏mi! )錛屽緱鍒?0縐?br>
涓轟簡渚夸簬璁＄畻br錛屽紩鍏ュ嚱鏁?/p>
g(x)= (1+x+x2/2!+x3/3!+…+xn1/n1!) (1+x+x2/2!+x3/3!+…+xn2/n2!)… (1+x+x2/2!+x3/3!+…+xnk/nk!)=鈭?(br*xr/r!)
g(x)縐頒負{br}鐨勬寚鏁版瘝鍑芥暟銆俠r=鈭? r! / ∏mi! )
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