The curriculum's permutation

Question

Consider the word CURRICULUM.How can i find number of ways in which 5 letter words can be formed using the letyers from the word CURRICULUM if each 5 letter word has exactly 3 different letters?My try: Arranging letters into groups.(CC)(UUU)(RR)(I)(M)(L).Case 1:-2 different ,other 3same letters 3*5!/3!.Is there any other case?

Answer 1

As user92774 said in the comments, the answer is $3320$. Here is the explanation for the seemingly random derivation.

The technique uses something called "Generating functions." If you were to expand that polynomial, the coefficient of $x^n$ would refer to the number of words one can make of size $n$ with the given letters. Notice that there are $10$ letters, and the largest coefficient is $10$.

Think of multiplying out the polynomial as "picking" one term inside each pair of parantheses, multiplying them together, and repeating for every permutation. Each permutation refers to a case of picking some number of $C, U, R, I, L, M$, and making a word out of it. The polynomials $(x+1)^3$ refer to the letters $I, L,$ and $M$ as there are only one of them. As you multiply them out, if you "pick" $x$ you choose one letter to appear in the word, if you "pick" $1$ then you choose the letter to not appear in the word. Similarly, $(1+x+\frac{x^2}{2})^2$ refers to $C$ and $R$. If you were to pick $\frac{x^2}{2}$, then there would be two $C$'s / $R$'s in the word. The reason why now it is divided by $2$ (more precisely, by $2!$) is that, since the same letter is repeated twice throughout the word, we will be counting the same permutations twice, so we divide by two.

Finally, the coefficient of $x^5$ is found because that refers to words with a size of $5$. Going in line with what I said previously, if we "picked" letters such that we had $5$ letters, then the coefficients, when multiplied together, would add up to $5$.

Answer 2

The possible unordered words are:

CCIRR, CCIUU, CCLRR, CCLUU, CCMRR, CCMUU, CCRRU, CCRUU, CIUUU, CLUUU, CMUUU, CRRUU, CRUUU, ILUUU, IMUUU, IRRUU, IRUUU, LMUUU, LRRUU, LRUUU, MRRUU, MRUUU

in alphabetical order.

Given an unordered word, the number of ordered words is given by a multinomial coefficient: $\binom{5}{2,1,2}=30$ for CCIRR, and so on.

If we do the bookkeeping, we obtain $560$ possible words.

Since this contradicts the previous answer/comment, I list the $560$ possibilities below:

30: CCIRR, CCRIR, CCRRI, CICRR, CIRCR, CIRRC, CRCIR, CRCRI, CRICR, CRIRC, CRRCI, CRRIC, ICCRR, ICRCR, ICRRC, IRCCR, IRCRC, IRRCC, RCCIR, RCCRI, RCICR, RCIRC, RCRCI, RCRIC, RICCR, RICRC, RIRCC, RRCCI, RRCIC, RRICC
30: CCIUU, CCUIU, CCUUI, CICUU, CIUCU, CIUUC, CUCIU, CUCUI, CUICU, CUIUC, CUUCI, CUUIC, ICCUU, ICUCU, ICUUC, IUCCU, IUCUC, IUUCC, UCCIU, UCCUI, UCICU, UCIUC, UCUCI, UCUIC, UICCU, UICUC, UIUCC, UUCCI, UUCIC, UUICC
30: CCLRR, CCRLR, CCRRL, CLCRR, CLRCR, CLRRC, CRCLR, CRCRL, CRLCR, CRLRC, CRRCL, CRRLC, LCCRR, LCRCR, LCRRC, LRCCR, LRCRC, LRRCC, RCCLR, RCCRL, RCLCR, RCLRC, RCRCL, RCRLC, RLCCR, RLCRC, RLRCC, RRCCL, RRCLC, RRLCC
30: CCLUU, CCULU, CCUUL, CLCUU, CLUCU, CLUUC, CUCLU, CUCUL, CULCU, CULUC, CUUCL, CUULC, LCCUU, LCUCU, LCUUC, LUCCU, LUCUC, LUUCC, UCCLU, UCCUL, UCLCU, UCLUC, UCUCL, UCULC, ULCCU, ULCUC, ULUCC, UUCCL, UUCLC, UULCC
30: CCMRR, CCRMR, CCRRM, CMCRR, CMRCR, CMRRC, CRCMR, CRCRM, CRMCR, CRMRC, CRRCM, CRRMC, MCCRR, MCRCR, MCRRC, MRCCR, MRCRC, MRRCC, RCCMR, RCCRM, RCMCR, RCMRC, RCRCM, RCRMC, RMCCR, RMCRC, RMRCC, RRCCM, RRCMC, RRMCC
30: CCMUU, CCUMU, CCUUM, CMCUU, CMUCU, CMUUC, CUCMU, CUCUM, CUMCU, CUMUC, CUUCM, CUUMC, MCCUU, MCUCU, MCUUC, MUCCU, MUCUC, MUUCC, UCCMU, UCCUM, UCMCU, UCMUC, UCUCM, UCUMC, UMCCU, UMCUC, UMUCC, UUCCM, UUCMC, UUMCC
30: CCRRU, CCRUR, CCURR, CRCRU, CRCUR, CRRCU, CRRUC, CRUCR, CRURC, CUCRR, CURCR, CURRC, RCCRU, RCCUR, RCRCU, RCRUC, RCUCR, RCURC, RRCCU, RRCUC, RRUCC, RUCCR, RUCRC, RURCC, UCCRR, UCRCR, UCRRC, URCCR, URCRC, URRCC
30: CCRUU, CCURU, CCUUR, CRCUU, CRUCU, CRUUC, CUCRU, CUCUR, CURCU, CURUC, CUUCR, CUURC, RCCUU, RCUCU, RCUUC, RUCCU, RUCUC, RUUCC, UCCRU, UCCUR, UCRCU, UCRUC, UCUCR, UCURC, URCCU, URCUC, URUCC, UUCCR, UUCRC, UURCC
20: CIUUU, CUIUU, CUUIU, CUUUI, ICUUU, IUCUU, IUUCU, IUUUC, UCIUU, UCUIU, UCUUI, UICUU, UIUCU, UIUUC, UUCIU, UUCUI, UUICU, UUIUC, UUUCI, UUUIC
20: CLUUU, CULUU, CUULU, CUUUL, LCUUU, LUCUU, LUUCU, LUUUC, UCLUU, UCULU, UCUUL, ULCUU, ULUCU, ULUUC, UUCLU, UUCUL, UULCU, UULUC, UUUCL, UUULC
20: CMUUU, CUMUU, CUUMU, CUUUM, MCUUU, MUCUU, MUUCU, MUUUC, UCMUU, UCUMU, UCUUM, UMCUU, UMUCU, UMUUC, UUCMU, UUCUM, UUMCU, UUMUC, UUUCM, UUUMC
30: CRRUU, CRURU, CRUUR, CURRU, CURUR, CUURR, RCRUU, RCURU, RCUUR, RRCUU, RRUCU, RRUUC, RUCRU, RUCUR, RURCU, RURUC, RUUCR, RUURC, UCRRU, UCRUR, UCURR, URCRU, URCUR, URRCU, URRUC, URUCR, URURC, UUCRR, UURCR, UURRC
20: CRUUU, CURUU, CUURU, CUUUR, RCUUU, RUCUU, RUUCU, RUUUC, UCRUU, UCURU, UCUUR, URCUU, URUCU, URUUC, UUCRU, UUCUR, UURCU, UURUC, UUUCR, UUURC
20: ILUUU, IULUU, IUULU, IUUUL, LIUUU, LUIUU, LUUIU, LUUUI, UILUU, UIULU, UIUUL, ULIUU, ULUIU, ULUUI, UUILU, UUIUL, UULIU, UULUI, UUUIL, UUULI
20: IMUUU, IUMUU, IUUMU, IUUUM, MIUUU, MUIUU, MUUIU, MUUUI, UIMUU, UIUMU, UIUUM, UMIUU, UMUIU, UMUUI, UUIMU, UUIUM, UUMIU, UUMUI, UUUIM, UUUMI
30: IRRUU, IRURU, IRUUR, IURRU, IURUR, IUURR, RIRUU, RIURU, RIUUR, RRIUU, RRUIU, RRUUI, RUIRU, RUIUR, RURIU, RURUI, RUUIR, RUURI, UIRRU, UIRUR, UIURR, URIRU, URIUR, URRIU, URRUI, URUIR, URURI, UUIRR, UURIR, UURRI
20: IRUUU, IURUU, IUURU, IUUUR, RIUUU, RUIUU, RUUIU, RUUUI, UIRUU, UIURU, UIUUR, URIUU, URUIU, URUUI, UUIRU, UUIUR, UURIU, UURUI, UUUIR, UUURI
20: LMUUU, LUMUU, LUUMU, LUUUM, MLUUU, MULUU, MUULU, MUUUL, ULMUU, ULUMU, ULUUM, UMLUU, UMULU, UMUUL, UULMU, UULUM, UUMLU, UUMUL, UUULM, UUUML
30: LRRUU, LRURU, LRUUR, LURRU, LURUR, LUURR, RLRUU, RLURU, RLUUR, RRLUU, RRULU, RRUUL, RULRU, RULUR, RURLU, RURUL, RUULR, RUURL, ULRRU, ULRUR, ULURR, URLRU, URLUR, URRLU, URRUL, URULR, URURL, UULRR, UURLR, UURRL
20: LRUUU, LURUU, LUURU, LUUUR, RLUUU, RULUU, RUULU, RUUUL, ULRUU, ULURU, ULUUR, URLUU, URULU, URUUL, UULRU, UULUR, UURLU, UURUL, UUULR, UUURL
30: MRRUU, MRURU, MRUUR, MURRU, MURUR, MUURR, RMRUU, RMURU, RMUUR, RRMUU, RRUMU, RRUUM, RUMRU, RUMUR, RURMU, RURUM, RUUMR, RUURM, UMRRU, UMRUR, UMURR, URMRU, URMUR, URRMU, URRUM, URUMR, URURM, UUMRR, UURMR, UURRM
20: MRUUU, MURUU, MUURU, MUUUR, RMUUU, RUMUU, RUUMU, RUUUM, UMRUU, UMURU, UMUUR, URMUU, URUMU, URUUM, UUMRU, UUMUR, UURMU, UURUM, UUUMR, UUURM

These were computed using GAP.