PERMUTATIONS - How to form permutations of n given things?

How to form permutations of n given things?								Counting permutations				no.of permutations		nPn=n!
1											1	1		1!
Insert new item first to the left of the previous permutations; and then shift its (new item's) place to the right by 1 position to get new permutations
2	1			i.e.	2	1					2 X 1	2		2!
2 shifts to the right by 1 position
	1	2			1	2
3 in left
3	2	1						2		3 x 2	3 x 2 x 1	6		3!
3	1	2
3 shifts to the right by 1 position
2	3	1						2
1	3	2
3 shifts to the right by 1 position
2	1	3						2
1	2	3
4 in the left most position
4	3	2	1					6		4 x 6	4 x 3 x 2 x 1	24		4!
4	3	1	2
4	2	3	1
4	1	3	2
4	2	1	3
4	1	2	3
4 shifts to the right by 1 position
3	4	2	1					6
3	4	1	2
2	4	3	1
1	4	3	2
2	4	1	3
1	4	2	3
4 shifts to the right by 1 position
3	2	4	1					6
3	1	4	2
2	3	4	1
1	3	4	2
2	1	4	3
1	2	4	3
4 shifts to the right by 1 position
3	2	1	4					6
3	1	2	4
2	3	1	4
1	3	2	4
2	1	3	4
1	2	3	4
and so on…..														5!

An Observation

An observation: If you add 3 numbers, make it's cube. Subtract from it individual cubes of these 3 numbers the answer of subtraction will always be divisible by 3.

Cuberoots of cubes of 10 to 99 in 15 seconds!

How to quickly find cube root?
Tell your friend to take any 2 digit number and calculate its cube. Tell him to tell you the answer I.e. give you the perfect cube number.
Step 1:Note the last (unit place) digit. If it is 0,1,4,5,6 or 9 the unit place digit of cube root is the last digit itself. If it is 2,3,7 or 8 the last digit of the cube root is 8,7,3 or 2 respectively.
Step 2:Leave aside the last 3 digits of the perfect cube from right.
Step 3: Consider the remaining number. This number will be maximum a 3 digit number. (Learn by heart cubes of 0 to 9). Find the largest perfect cube smaller than this considered number.
Step 4: The cube root of this largest perfect cube is the tens place digit of the cube root of the number given by your friend.
Step 5: write step 4's answer. To its right write answer of steps 1.
Step 6: This is the answer. I.e. the 2 digit number originally taken by your friend. I mean the cube root of the number told (given) by your friend.

Arithmatic Progression - Triplets whose squares form an A.P.

p	d1	p+md1	p+(m+n)d1	p2	(p+md1)2	[p+(m+n)d1]2	t2-t1	t3-t2
1	1	5	7	1	25	49	24	24
1	4	29	41	1	841	1681	840	840
3	2	15	21	9	225	441	216	216
5	4	145	205	25	21025	42025	21000	21000
7	2	73	103	49	5329	10609	5280	5280
7	1	35	49	49	1225	2401	1176	1176
7	2	13	17	49	169	289	120	120
7	2	17	23	49	289	529	240	240
15	2	75	105	225	5625	11025	5400	5400
17	4	137	193	289	18769	37249	18480	18480
17	9	305	431	289	93025	185761	92736	92736
17	2	25	31	289	625	961	336	336
31	3	109	151	961	11881	22801	10920	10920
31	38	1285	1817	961	1651225	3301489	1650264	1650264
35	2	85	115	1225	7225	13225	6000	6000
41	2	89	119	1681	7921	14161	6240	6240
41	2	85	113	1681	7225	12769	5544	5544
47	1	65	79	2209	4225	6241	2016	2016
49	2	91	119	2401	8281	14161	5880	5880
51	4	159	219	2601	25281	47961	22680	22680
69	2	111	141	4761	12321	19881	7560	7560
73	5	193	263	5329	37249	69169	31920	31920
85	2	125	155	7225	15625	24025	8400	8400
89	38	1381	1951	7921	1907161	3806401	1899240	1899240
89	3	149	191	7921	22201	36481	14280	14280
97	16	593	833	9409	351649	693889	342240	342240
97	1	113	127	9409	12769	16129	3360	3360
103	34	1157	1633	10609	1338649	2666689	1328040	1328040
127	14	533	743	16129	284089	552049	267960	267960
147	2	183	213	21609	33489	45369	11880	11880
161	34	1249	1759	25921	1560001	3094081	1534080	1534080
167	30	1037	1457	27889	1075369	2122849	1047480	1047480
193	8	377	497	37249	142129	247009	104880	104880
217	4	293	353	47089	85849	124609	38760	38760
223	2	257	287	49729	66049	82369	16320	16320
223	26	925	1289	49729	855625	1661521	805896	805896
271	22	821	1129	73441	674041	1274641	600600	600600
281	26	1009	1399	78961	1018081	1957201	939120	939120
311	18	725	977	96721	525625	954529	428904	428904
329	22	901	1231	108241	811801	1515361	703560	703560
367	10	557	697	134689	310249	485809	175560	175560
383	6	485	569	146689	235225	323761	88536	88536
391	2	421	449	152881	177241	201601	24360	24360
401	14	709	919	160801	502681	844561	341880	341880
449	2	481	511	201601	231361	261121	29760	29760