How many numbers with a maximum of 5 digits, greater than 4000, can be formed with the digits 2, 3, 4, 5, 6; if repetition is allowed for 2 and 3 only?


Question and Answers Forum

All Questions Topic List
Permutation and Combination Questions
Previous in All Question Next in All Question
Previous in Permutation and Combination Next in Permutation and Combination
Question Number 198576 by mr W last updated on 22/Oct/23

How many numbers with a maximum of 5 digits, greater than 4000, can be formed with the digits 2, 3, 4, 5, 6; if repetition is allowed for 2 and 3 only?
Commented by mr W last updated on 22/Oct/23

$Q 198242 r e p o s t e d f o r a l t e r n a t i v e$ $s o l u t i o n s$
Commented by MM42 last updated on 22/Oct/23

$S i r W$ $y o u a r e r i g h t . i w a s n o t c a r e f u l .$ $t h a n k y o u f o r y o u r c o n s i d e r a t i o n .$
Commented by mr W last updated on 22/Oct/23

$y o u a r e w e l c o m e s i r!$
Commented by nikif99 last updated on 23/Oct/23
$A more complicated solution I. four digit numbers determinant (((1st),( 2nd),( 3rd),( 4th),( cases^★ ))) 4or5or6^★ { ((2or3 { ((2or3 { ((2or3 (3×2×2×2=24))),((5or6 (24))) :})),((5or6 { ((2or3 (24))),((6 (12))) :})) :})),((5or6^★ { ((2or3 { ((2or3 (24))),((6^★ (12))) :})),((6 { ((2or3 (12))),(( × (0))) :})) :})) :} ★: the product of all possible cases through a routing of digits. Subtotal 132 II. five digit numbers determinant (((1st),( 2nd),( 3rd),( 4th),( 5th),( cases))) determinant (((2or3 { ((2or3 { ((2or3 { ((2or3 { ((2or3 (32))),((4or5or6 (48))) :})),((4or5or6 { ((2or3 (48))),((5or6 (48))) :})) :})),((4or5or6 { ((2or3 { ((2or3 (48))),((5or6 (48))) :})),((5or6 { ((2or3 (48))),((6 (24))) :})) :})) :})),((4or5or6^★ { ((2or3 { ((2or3 { ((2or3 (48))),((5or6 (48))) :})),((5or6 { ((2or3 (48))),((6 (24))) :})) :})),((5or6^★ { ((2or3 { ((2or3 (48))),((6^★ (24))) :})),((6 { ((2 (12))),((3 (12))) :})) :})) :})) :}))) Subtotal 608 4or5or6^★ { ((2or3 { ((2or3 { ((2or3 { ((2or3 (48))),((5or6 (48))) :})),((5or6 { ((2or3 (48))),((6 (24))) :})) :})),((5or6 { ((2or3 { ((2or3 (48))),((6 (24))) :})),((6 { ((2 (12))),((3 (12))) :})) :})) :})),((5or6^★ { ((2or3 { ((2or3 { ((2or3 (48))),((6 (24))) :})),((6^★ { ((2 (8))),((3 (8))) :})) :})),((6 { ((2 { ((2 (8))),((3 (8))) :})),((3 { ((2 (8))),((3 (8))) :})) :})) :})) :} Subtotal 384 Total 1124 ★: if for exemple 4 is selected initialy, then one of {5,6} is selected next, and finally the last one.$
$A m o r e c o m p l i c a t e d s o l u t i o n$ $I . f o u r d i g i t n u m b e r s$ $\begin{matrix} 1 s t & 2 n d & 3 r d & 4 t h & {c a s e s}^{★} \end{matrix}$ $4 o r 5 o r 6^{★} {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 (3 \times 2 \times 2 \times 2 = 24) \\ 5 o r 6 (24) \end{cases} \\ 5 o r 6 {\begin{cases} 2 o r 3 (24) \\ 6 (12) \end{cases} \end{cases} \\ 5 o r 6^{★} {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 (24) \\ 6^{★} (12) \end{cases} \\ 6 {\begin{cases} 2 o r 3 (12) \\ \times (0) \end{cases} \end{cases} \end{cases}$ $★ : t h e p r o d u c t o f a l l p o s s i b l e c a s e s t h r o u g h a r o u t i n g o f d i g i t s .$ $S u b t o t a l 132$ $I I . f i v e d i g i t n u m b e r s$ $\begin{matrix} 1 s t & 2 n d & 3 r d & 4 t h & 5 t h & c a s e s \end{matrix}$ $\begin{matrix} 2 o r 3 {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 (32) \\ 4 o r 5 o r 6 (48) \end{cases} \\ 4 o r 5 o r 6 {\begin{cases} 2 o r 3 (48) \\ 5 o r 6 (48) \end{cases} \end{cases} \\ 4 o r 5 o r 6 {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 (48) \\ 5 o r 6 (48) \end{cases} \\ 5 o r 6 {\begin{cases} 2 o r 3 (48) \\ 6 (24) \end{cases} \end{cases} \end{cases} \\ 4 o r 5 o r 6^{★} {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 (48) \\ 5 o r 6 (48) \end{cases} \\ 5 o r 6 {\begin{cases} 2 o r 3 (48) \\ 6 (24) \end{cases} \end{cases} \\ 5 o r 6^{★} {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 (48) \\ 6^{★} (24) \end{cases} \\ 6 {\begin{cases} 2 (12) \\ 3 (12) \end{cases} \end{cases} \end{cases} \end{cases} \end{matrix}$ $S u b t o t a l 608$ $4 o r 5 o r 6^{★} {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 (48) \\ 5 o r 6 (48) \end{cases} \\ 5 o r 6 {\begin{cases} 2 o r 3 (48) \\ 6 (24) \end{cases} \end{cases} \\ 5 o r 6 {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 (48) \\ 6 (24) \end{cases} \\ 6 {\begin{cases} 2 (12) \\ 3 (12) \end{cases} \end{cases} \end{cases} \\ 5 o r 6^{★} {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 {\begin{cases} 2 o r 3 (48) \\ 6 (24) \end{cases} \\ 6^{★} {\begin{cases} 2 (8) \\ 3 (8) \end{cases} \end{cases} \\ 6 {\begin{cases} 2 {\begin{cases} 2 (8) \\ 3 (8) \end{cases} \\ 3 {\begin{cases} 2 (8) \\ 3 (8) \end{cases} \end{cases} \end{cases} \end{cases}$ $S u b t o t a l 384$ $T o t a l 1124$ $★ : i f f o r e x e m p l e 4 i s s e l e c t e d i n i t i a l y, t h e n o n e o f {5, 6} i s s e l e c t e d n e x t,$ $a n d f i n a l l y t h e l a s t o n e .$
Commented by mr W last updated on 23/Oct/23

$m e t h o d u s i n g g e n e r a t i n g f u n c t i o n$ $s e e m s t o b e c a m p l i c a t e d a t f i r s t$ $g l a n c e, b u t a c t u a l l y t h i s s o l u t i o n$ $m e t h o d i s g e n e r a l a n d t h u s v e r y$ $p o w e r f u l . t r y t o f i n d t h e n u m b e r$ $o f 7 - d i g i t n u m b e r s w i t h e x a c t l y$ $t h e s a m e c o n d i t i o n s . c a n y o u g i v e$ $a s o l u t i o n f o r t h i s c a s e ?$
Commented by mr W last updated on 23/Oct/23

Commented by nikif99 last updated on 23/Oct/23

$I t i s q u i t e d i f f i c u l t t o m a n a g e m a n u a l l y$ $s u c h a t r e e w i t h 7 l e v e l s .$ $T r y i n g a n o t h e r a p p r o a c h, I u s e d$ $s e l f - d e v e l o p p e d s o f t w a r e v i a F o r t r a n$ $a n d I f o u n d 8.864 c a s e s f o r 7 - d i g i t$ $n u m b e r s (a n d 3.040 f o r 6 - d i g i t) .$ $P . S . I d o n o t k n o w w h a t o t h e r m e m b e r s$ $s e e o n m y s o l u t i o n c o m m e n t;$ $I p e r s o n a l l y s e e o n l y t h e t i t l e i n b o l d$ $(‘ ‘ 4 - d i g i t {n u m b e r s}^{″}) w i t h o u t a n a l y s i s$ $w i t h o p e n i n g c u r l y b r a c e s .$
Commented by mr W last updated on 23/Oct/23

$8864 n u m b e r s w i t h 7 d i g i t s, t h i s i s$ $c o r r e c t, t h a n k s a l o t! i a l s o g o t t h i s$ $r e s u l t, v e r y e a s i l y w i t h m y m e t h o d$ $a p p l y i n g g e n e r a t i n g f u n c t i o n . t h e$ $r e s u l t i s t h e c o e f f i c i e n t o f x^{7} t e r m$ $i n 7! e^{2 x} {(1 + x)}^{3}, w h i c h i s 8864 .$ $s i m i l a r l y f o r 6 - d i g i t n u m b e r s i t i s$ $t h e c o e f f i c i e n t o f x^{6} t e r m i n$ $6! e^{2 x} {(1 + x)}^{3}, w h i c h i s 3040 .$
Commented by nikif99 last updated on 23/Oct/23

Commented by mr W last updated on 23/Oct/23

$n i c e p r e p a r e d!$
	Answered by mr W last updated on 23/Oct/23

	$2 a n d 3 c a n b e u s e d z e r o t i m e, o n e$ $t i m e o r a n y t i m e s \Rightarrow (1 + x + x^{2} + x^{3} + . . .)$ $4, 5 a n d 6 c a n o n l y b e u s e d z e r o t i m e$ $o r o n e t i m e \Rightarrow (1 + x)$ $\underset{―}{5 - d i g i t n u m b e r s :}$ $n u m b e r o f s u c h n u m b e r s i s t h e$ $c o e f f i c i e n t o f x^{5} t e r m i n$ $5! {(1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + . . .)}^{2} {(1 + x)}^{3}$ $= 5! e^{2 x} {(1 + x)}^{3}$ $w h i c h i s 992 .$ $t h a t m e a n s w e c a n f o r m 992 s u c h$ $5 - d i g i t n u m b e r s .$
	Commented by Tawa11 last updated on 23/Oct/23

	$Thanks sir .$ $I appreciate your time, God bless you sir .$
	Commented by mr W last updated on 22/Oct/23

	Commented by mr W last updated on 22/Oct/23

	$\underset{―}{4 - d i g i t n u m b e r s :}$ $t h e f i r s t d i g i t m u s t b e 4, 5 o r 6 .$ $w e s e l e c t a t f i r s t t h e f i r s t d i g i t,$ $t h e r e a r e 3 w a y s . t h e n w e f o r m$ $3 - d i g i t n u m b e r s w i t h t h e o t h e r 4$ $d i g i t s . s i m i l a r l y t o a b o v e, t h e n u m b e r$ $o f s u c h n u m b e r s i s t h e c o e f f i c i e n t$ $o f x^{3} t e r m i n 3! e^{2 x} {(1 + x)}^{2}, w h i c h i s$ $44 . s o t o t a l l y w e c a n f o r m$ $3 \times 44 = 132 v a l i d 4 - d i g i t n u m b e r s .$
	Commented by mr W last updated on 22/Oct/23

	Commented by mr W last updated on 22/Oct/23

	$a l l t o g e t h e r : 992 + 132 = 1124 ✓$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum