Tiêu đề Permutations & Combinations Engg Practice Sheet Solution (HSC 26).pdf ✅

web ̈vm I mgv‡ek  Engineering Practice Sheet Solution (HSC 26) 1 05 web ̈vm I mgv‡ek Permutations & Combinations WRITTEN weMZ mv‡j BUET-G Avmv cÖkœvejx 1| „HIPPOPOTAMUS‟ kãwUi eY© ̧‡jv †_‡K 1wU ̄^ieY© I 2wU e ̈ÄbeY© wb‡q KZ ̧‡jv kã MVb Kiv m¤¢e †hb ̄^ieY©wU memgq gvSLv‡b _v‡K| [BUET 20-21] mgvavb: HIPPOPOTAMUS kãwU‡ZÑ (i) wfbœ e ̈ÄbeY© Av‡Q (7 – 3 + 1) = 5wU (H, P, T, M, S) Ges GKB e ̈ÄbeY© P Av‡Q 3wU| (ii) wfbœ ̄^ieY© Av‡Q (5 – 2 + 1) = 4wU (I, O, A, U) Ges GKB ̄^ieY© O Av‡Q 2wU| 1wU ̄^ieY© I 2wU e ̈ÄbeY© wb‡q MwVZ msL ̈v †hb ̄^ieY© memgq gvSLv‡b _v‡KÑ Case-1: 2wU e ̈ÄbeY©(GKB), 1wU ̄^ieY© 1C1  4C1  2! 2! = 4 Case-2: 2wU e ̈ÄbeY©(wfbœ), 1wU ̄^ieY© 5C2  4C1  2! = 80 †gvU kã msL ̈v = (80 + 4) = 84wU (Ans.) 2| 1, 2, 3,4, 5, 6, 7, 8 wPwýZ AvUwU KvD›Uvi †_‡K Kgc‡ÿ 1wU we‡Rvo I 1wU †Rvo KvD›Uvi wb‡q GKev‡i 4wU KvD›Uvi wb‡j mgv‡ek msL ̈v KZ n‡e? [BUET 19-20] mgvavb: we‡Rvo(4wU KvD›Uvi) †Rvo(4wU KvD›Uvi) mgv‡ek msL ̈v 1 3 4C1  4C3 = 16 2 2 4C2  4C2 = 36 3 1 4C3  4C1 = 16  †gvU mgv‡ek msL ̈v = 16 + 36 + 16 = 68wU (Ans.) 3| mvZwU eY© A, B, C, D, E, F, I G †K Ggbfv‡e mvRv‡Z n‡e †hb A Ges B eY©Øq KLbB cvkvcvwk bv _v‡K| KZ cÖKv‡i GB kZ© †g‡b eY© ̧‡jv‡K mvRv‡bv †h‡Z cv‡i? [BUET 18-19] mgvavb: A I B KY©Øq‡K cvkvcvwk †i‡L web ̈vm msL ̈v = (7 – 2 + 1)!  2! = 6!  2!  wb‡Y©q web ̈vm msL ̈v = 7! – 6!  2! = 3600 (Ans.) 4| EXAMINATION kãwUi e ̈ÄbeY© ̧‡jv‡K GK‡Î bv †i‡L KZ iK‡g mvRv‡bv hvq? [BUET 17-18] mgvavb: EXAMINATION kãwU‡Z: (i) †gvU eY© Av‡Q 11wU (GKB eY© A = 2wU, I = 2wU, N = 2wU) (ii) e ̈ÄbeY© Av‡Q 5wU (&GKB eY© N = 2wU)  wb‡Y©q web ̈vm msL ̈v = kãwUi web ̈vm msL ̈v-e ̈ÄbeY© ̧‡jv‡K GK‡Î †i‡L web ̈vm msL ̈v = 11! 2!2!2! – (11 – 5 + 1)! 2!  2!  5! 2! = 11! 2!2!2! – 7! 2!  2!  5! 2! = 4914000 (Ans.) 5| CALCULUS kãwUi eY© ̧‡jvi me ̧‡jv‡K GK‡Î wb‡q KZ cÖKv‡i mvRv‡bv hvq? GB web ̈vm ̧‡jvi KZ ̧‡jv‡Z cÖ_‡g I †k‡l GKB Aÿi _vK‡e? [BUET 16-17] mgvavb: CALCULUS kãwU‡ZÑ (i) †gvU eY© Av‡Q 8wU [Gi g‡a ̈ GKB eY© C = 2wU, U = 2wU, L = 2wU]  wb‡Y©q web ̈vm msL ̈v = 8! 2!2!2! = 5040 (Ans.) (ii) cÖ_‡g I †k‡l ‘C’ †i‡L web ̈vm msL ̈v = 6! 2!2! = 180 cÖ_g I †k‡l GKB Aÿi (C, L, U) ivLv hvq 3 fv‡e|  †gvU web ̈vm msL ̈v = 3  180 = 540 (Ans.) 6| 6 Rb MwYZ I 4 Rb c`v_©weÁv‡bi QvÎ †_‡K 6 R‡bi GKwU KwgwU MVb Ki‡Z n‡e| KwgwUwU KZ cÖKv‡i MVb Kiv †h‡Z cv‡i †hb MwY‡Zi Qv·`i msL ̈vMwiôZv _v‡K? [BUET 12-13] mgvavb: MwYZ wefvM(6 Rb) c`v_©weÁvb wefvM(4 Rb) KwgwU msL ̈v 6 0 6C6  4C0 = 1 5 1 6C5  4C1 = 24 4 2 6C4  4C2 = 90  KwgwU MVb Kiv †h‡Z cv‡i = (90 + 24 + 1) = 115 cÖKv‡i| (Ans.) 7| 6 Rb I 8 Rb †L‡jvqv‡oi `ywU `j †_‡K 11 Rb †L‡jvqv‡oi GKwU wμ‡KU wUg MVb Ki‡Z n‡e hv‡Z 6 R‡bi `j †_‡K AšÍZ 4 Rb †L‡jvqvo H wU‡g _v‡K| wμ‡KU wUgwU †gvU KZ cÖKv‡i MVb Kiv †h‡Z cv‡i? [BUET 11-12] mgvavb: 1g `j(6 Rb) 2q `j(8 Rb) `‡ji msL ̈v 4 7 6C4  8C7 = 120 5 6 6C5  8C6 = 168 6 5 6C6  8C5 = 56  wμ‡KU wUgwU MVb Kiv †h‡Z cv‡i = (120 + 168 + 56) = 344 Dcv‡q| (Ans.)
2  Higher Math 1st Paper Chapter-5 8| `yÕwU fv‡Mi cÖ‡Z ̈K fv‡M 5 wU K‡i †gvU 10wU cÖkœ n‡Z GKRb cixÿv_x©‡K 6wU cÖ‡kœi DËi w`‡Z n‡e| †Kv‡bv fvM †_‡K 4wUi †ewk cÖ‡kœi DËi Kiv wbwl×| H cixÿv_x© KZ Dcv‡q cÖkœ ̧‡jv evQvB Ki‡Z cvi‡e? [BUET 09-10] mgvavb: cÖ_g fvM(5wU cÖkœ) wØZxq fvM(5wU cÖkœ) cÖkœ ̧‡jv evQvB Kivi msL ̈v 4 2 5C4  5C2 = 50 3 3 5C3  5C3 = 100 2 4 5C2  5C4 = 50  cixÿv_©x cÖkœ ̧‡jv evQvB Ki‡Z cv‡i = (50 + 100 + 50) = 200 Dcv‡q (Ans.) weMZ mv‡j KUET-G Avmv cÖkœvejx 9| n msL ̈K wfbœ wfbœ e ̄‘ n‡Z h_vμ‡g (r + 1), (r + 2) Ges (r + 3) msL ̈K e ̄‘ wb‡q hZ ̧wj mgv‡ek MVb Kiv hvq Zv‡`i AbycvZ 15 : 24 : 28 n‡j n I r Gi gvb wbY©q Ki| [KUET 19-20] mgvavb: nCr + 1 : nCr + 2 = 15 : 24  n! (r + 1)! (n – r – 1)!  (r + 2)! (n – r – 2)! n! = 15 24 = 5 8  r + 2 n – r – 1 = 5 8  5n – 5r – 5 = 8r + 16  5n – 13r = 21 ....... (i) nCr + 2 : nCr + 3 = 24 : 28  n! (r + 2)! (n – r – 2)!  (r + 3)! (n – r – 3)! n! = 24 28 = 6 7  r + 3 n – r – 2 = 6 7  6n – 6r – 12 = 7r + 21  6n – 13r = 33 ......... (ii) (i) I (ii) mgvavb K‡i, n = 12, r = 3 (Ans.) 10| 15 Rb Qv‡Îi ga ̈ †_‡K cÖwZ KwgwU‡Z 5 Rb wn‡m‡e †gvU 3 wU KwgwU MVb Ki‡Z n‡e| KZ Dcv‡q H KwgwU ̧‡jv MVb Kiv hv‡e? [KUET 05-06] mgvavb: 15 Rb †_‡K 5 Rb‡K wb‡q KwgwU MVb Kivi Dcvq = 15C5 evwK (15 – 5) ev 10 Rb †_‡K 5 Rb wb‡q KwgwU MVb Kivi Dcvq = 10C5 Ges (15 – 10) ev 5 Rb †_‡K 5 Rb wb‡q KwgwU MVb Kivi Dcvq = 5C5  wZbwU KwgwU MVb Kivi Dcvq = 15C5  10C5  5C5 3! = 126126 (Ans.) 11| GKRb ms‡KZ Kvi‡Ki PqwU cZvKv Av‡Q hv‡`i g‡` ̈ GKwU mv`v, `ywU meyR Ges wZbwU jvj; †m (i) GK m‡1⁄2 QqwU cZvKv e ̈envi K‡i (ii) GK m‡1⁄2 cvuPwU e ̈envi K‡i KZwU wewfbœ ms‡KZ cv‡e? [KUET 04-05] mgvavb: (i) 6! 2! 3! = 60 (Ans.) (ii) W G R ms‡KZ 1 2 2 5! 2!2! = 30 1 1 3 5! 3! = 20 0 2 3 5! 3!2! = 10  †gvU = 60 (Ans.) 12| „ENGINEERING‟ kãwUi meKqwU eY©‡K KZ wewfbœ iK‡g mvRv‡bv hvq Zv wbY©q Ki| Zv‡`i KZ ̧‡jv‡Z e wZbwU GK‡Î ̄’vb `Lj Ki‡e Ges KZ ̧‡jv‡Z Giv cÖ_g ̄’vb `Lj Ki‡e? [KUET 03-04; RUET 08-09, 03-04] mgvavb: mvRv‡bv msL ̈v = 11! 3! 3! 2! 2! (Ans.) E wZbwU‡K GK‡Î ivL‡j, mvRv‡bvi Dcvq = 9! 3! 2! 2! (Ans.) E wZbwU‡K hw` 1g ̄’vb `Lj K‡i Z‡e mvRv‡bv msL ̈v = 8! 3! 2! 2! (Ans.) 13| 6 Rb MwYZ I 4 Rb c`v_©weÁv‡bi QvÎ †_‡K 6 R‡bi GKwU KwgwU MVb Ki‡Z n‡e| KwgwUwU KZ cÖKv‡i MVb Kiv †h‡Z cv‡i †hb MwY‡Zi Qv·`i msL ̈vMwiôZv _v‡K? [KUET 03-04; BUET 12-13] mgvavb: MwYZ wefvM(6 Rb) c`v_©weÁvb wefvM(4 Rb) KwgwU msL ̈v 6 0 6C6  4C0 = 1 5 1 6C5  4C1 = 24 4 2 6C4  4C2 = 90  KwgwU MVb Kiv †h‡Z cv‡i = (90 + 24 + 1) = 115 cÖKv‡i| (Ans.) weMZ mv‡j RUET-G Avmv cÖkœvejx 14| 7 Rb cyiæl I 6 Rb gwnjvi ga ̈ n‡Z 5 m`‡m ̈i GKwU KwgwU KZ fv‡e •Zwi Kiv hv‡e hv‡Z Kgc‡ÿ 3 Rb cyiæl _vK‡e? [RUET 19-20] mgvavb: cyiæl(7 Rb) gwnjv(6 Rb) Dcvq msL ̈v 4 2 7C3  6C2 = 525 7 1 7C4  6C1 = 210 5 0 7C5  6C0 = 21  KwgwU MVb Kivi Dcvq = 525 + 210 + 21 = 756 Dcv‡q| (Ans.) 15| 1, 2, 3, 4, 5, 6, 7 AsK ̧‡jv GKevi gvÎ e ̈envi K‡i MwVZ I 5 Øviv AwefvR ̈ 7 AsKwewkó msL ̈v ̧‡jv gv‡bi EaŸ©μgvbymv‡i mvRv‡bv nj| D3 ZvwjKvq 2000 Zg msL ̈vwU KZ? [RUET 15-16] mgvavb: ïiæ‡Z 1 I †k‡l 5 ev‡` Ab ̈ †h‡Kv‡bv msL ̈v †i‡L web ̈vm: 1 5 4 3 2 1 5  (2, 3, 4, 6, 7)  (1)
web ̈vm I mgv‡ek  Engineering Practice Sheet Solution (HSC 26) 3  cÖ_‡g 1 wewkó 5 Øviv AwefvR ̈ msL ̈v, = 1  5  4  3  2  1  5 = 600wU A_©vr, msL ̈vwUi cÖ_g A1⁄4 4 wØZxq Ae ̄’v‡b 1 †i‡L web ̈vm: 1 1 4 3 2 1 4  (2, 3, 6, 7)  (4)  (1)  cÖ_‡g 4 I wØZxq Ae ̄’v‡b 1 †i‡L web ̈vm, = 1  1  4  3  2  1  4 = 96wU  cÖ_‡g 4 I wØZxq Ae ̄’v‡b 1 A_ev 2 †i‡L web ̈vm = 96  2 = 192 A_©vr, msL ̈vwUi wØZxq A1⁄4 3 Z...Zxq Ae ̄’v‡b 1 †i‡L web ̈vm: 1 1 1 3 2 1 3  (2, 6, 7)  (4)  (3)  (1)  cÖ_‡g 4, wØZxq 3 Ges Z...Zxq Ae ̄’v‡b 1 †i‡L web ̈vm = 1  1  1  3  2  1  3 = 18wU  431 w`‡q ïiæ Ggb msL ̈v (1800 + 192 + 18)wU = 2010wU  431 w`‡q ïiæ Ggb 8Zg msL ̈vwUB wb‡Y©q 2000 Zg msL ̈v 4312567 4315267 4312576 4315276 4312657 4315627 4312675 4315672  wb‡Y©q e„nËg msL ̈v = 4315672 (Ans.) 16| PERMUTATIONS kãwUi eY© ̧‡jv †_‡K GKwU ̄^ieY© Ges 2wU e ̈ÄbeY© wb‡q KZ ̧‡jv kã MVb Kiv hvq, †hb ̄^ieY©wU memgq gvSLv‡b _v‡K? [RUET 12-13] mgvavb: PERMUTATIONS kãwU‡ZÑ (i) wfbœ e ̈ÄbeY© Av‡Q (7 – 2 + 1) = 6wU (P, R, M, N, S, T) GKB e ̈Äb eY© T Av‡Q 2wU (ii) wfbœ ̄^ieY© Av‡Q 5wU (A, E, I, O, U) 1wU ̄^ieY© I 2wU e ̈ÄbeY© wb‡q MwVZ msL ̈v †hb ̄^ieY© memgq gvSLv‡b _v‡KÑ Case-1: 2wU e ̈ÄbeY©(GKB), 1wU ̄^ieY© 2C2  5C1  2! 2! = 5 Case-2: 2wU e ̈ÄbeY©(wfbœ), 1wU ̄^ieY© 6C2  5C1  2! = 150 †gvU kã msL ̈v = (5 + 150) = 155wU (Ans.) 17| ENGINEERING kãwU n‡Z cÖwZevi 4 wU K‡i Aÿi wb‡q KZ ̧‡jv kã MVb Kiv hv‡e? [RUET 11-12; 10-11] mgvavb: kã MV‡bi Dcvqmg~n wb¤œiƒc: ENGINEERING kãwU‡Z E-3; N-3; G-2; I-2; R-1 evQvB c×wZ web ̈vm msL ̈v Case-1: 3 wU GKB, GKwU wfbœ 2C1  4C1  4! 3! = 32 Case-2: 2 wU GKB, 2 wU GKB 4C2  4! 2! 2! = 36 Case-3: 2 wU GKB, 2 wU wfbœ 4C1  4C2  4! 2! = 288 Case-4: 4 wUB wfbœ 5 P4 = 120  †gvU Dcvq msL ̈v = 32 + 36 + 288 + 120 = 476 (Ans.) 18| „COMPUTER‟ k‡ãi Aÿi ̧‡jv n‡Z 3wU Aÿi wb‡q MwVZ kã msL ̈v wbY©q Ki hvi cÖ‡Z ̈KwU‡Z Kgc‡ÿ GKwU ̄^ieY© _v‡K| [RUET 09-10] mgvavb: ̄^ieY© Av‡Q Av‡Q 3 wU| e ̈ÄbeY© Av‡Q 5 wU|  Dchy3 kZ© †g‡b MwVZ kã msL ̈v 8 P3 – 5 P3 = 276 (Ans.) 19| „ENGINEERING‟ kãwUi meKqwU eY©‡K KZ wewfbœ iK‡g mvRv‡bv hvq Zv wbY©q Ki| Zv‡`i KZ ̧‡jv‡Z e wZbwU GK‡Î ̄’vb `Lj Ki‡e Ges KZ ̧‡jv‡Z Giv cÖ_g ̄’vb `Lj Ki‡e? [RUET 08-09, 03-04; KUET 03-04] mgvavb: mvRv‡bv msL ̈v = 11! 3! 3! 2! 2! (Ans.) E wZbwU‡K GK‡Î ivL‡j, mvRv‡bvi Dcvq = 9! 3! 2! 2! (Ans.) E wZbwU‡K hw` 1g ̄’vb `Lj K‡i Z‡e mvRv‡bv msL ̈v = 8! 3! 2! 2! (Ans.) 20| MATHEMATICS kãwUi meKqwU eY©‡K KZ wewfbœ iK‡g mvRv‡bv hvq Zv wbY©q Ki| ̄^ieY© ̧‡jv‡K GK‡Î †i‡L KZ cÖKv‡i mvRv‡bv hvq Zv wbY©q Ki| [RUET 07-08] mgvavb: ‘MATHEMATICS’ kãwU‡ZÑ (i) †gvU eY© Av‡Q 11wU (GKB eY© 2M, 2A, 2T) (ii) ̄^ieY© Av‡Q 4wU (GKB eY© 2A) Ges e ̈ÄbeY© Av‡Q 7wU (GKB eY© 2M, 2T)  meKqwU eY© wb‡q mvRv‡bvi Dcvq msL ̈v = 11! 2!2!2! (Ans.)  ̄^ieY© ̧‡jv GK‡Î †i‡L mvRv‡bv Dcvq msL ̈v = 8! 2!2!  4! 2! = 120960 (Ans.) 21| cÖgvY Ki †h, „Rajshahi‟ kãwUi Aÿi ̧‡jvi web ̈vm msL ̈v „Barisal‟ kãwUi Aÿi ̧‡jvi web ̈vm msL ̈vi Pvi ̧Y| [RUET 06-07] mgvavb: Rajshahi Gi web ̈vm msL ̈v = 8! 2! 2! = 10,080; Barisal Gi web ̈vm msL ̈v = 7! 2! = 2,520  Rajshahi Gi web ̈vm msL ̈v = 4  Barisal Gi web ̈vm msL ̈v|