Question

(3 points) How many integer solutions of + +33 +34 = 30 are there with I, > 0, 2 <5, 12 <11, 13 < 13, 14 < 23?

Answer 1

07 A.ccording to question we have to find how many is possible for integer solution (A) Hq t U + Ug + My =30 where Nizo, 4, 55, H₂E11 My < 23. уз < 13 ) # Before asolving, this question belong's from 6 combinations topic. Now, x + N2 + N₂ + xy = 30 De can be = 30 written in i=1 this form. where Hi > Here we i Meaning of 21+ x2 + x3 +Hy = 30% there are 8307 identical number (like lll. ly so time or El=30 - which can be distributed simong's 'Ky different May that ） -30 201 number.

Similar m formula Mit Ut isk Yui =n + kk = iil and Mizo then this is a case of 'n identical number ( like I, I, upto ..n.) which can be distributted to 'k) different number and the total number of distribution is calculated nak- by c Cx-l stk-1 'Cx-1 (+k-))! (1+k-1-(K-1))!(K-1)! = (1+k-1)! n! (K-1)! Do, now @ M + H₂ + H₂ t ly = 30 Ni 20 given in question H 25, H₂ Elly Uz £13 and Hy £ 23. (this is additional condition

pop let Change => 0_5-24 -) 5-34120 ilet 5-2, =P (plug this pealue = p zo 5-PE Exi Aimilarly H2 <1 = 117.HQ XO Let 11- = 0 again = q zo á K2= ll_q again N = 13 Lot 13 - 33 20 13 - Az = r again » r zo : N₂ = 13-9 again My <23 23 - My Let t = 23-lu t zo ... Wy: - 23-t. 느 & ny now W!) plug value of 7,) V3 where P, q, 7, t zo (positive).

- 5-P + 11-9 + 13-11+ 23-5 Mit Ng + H3 + Hy = 30 09 = 30 52- (P+9+9++) = 30 =) Ptq + rtt 52-30 22 Ly 8. P+9+977 + = again here p, q, r, t zo Here we read as "223 identical number which can be distributed to four different number (P, V, 9 → No, total numbersih inte garn solution is equal to total number of distribution of '22 identical number among "4) different number. therefore total number of integer solution ñ+k- 'Cx- - (n+k-1)! n!(k-1).

© Doy n= 221 k=u 22+ 4-1 total number of integer solution Curl (22+4-1)! (22)! (4-1)! こ 25! (22)! 3! 25x24x23x022) (22)! 3X2X1 = 25x24x 23 25 x 23 x 4 = 230 0 Answer There is 2300 solution is possible for 211+ H2 + x3 + Hy = 30, Hi 30, 31,55 My <13., Ny 5 23 May be helpful for you!! please uprote me!! Thank you!!

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