IMAGE is a 5 letters word. We have to arrange all 5 letters of word IMAGE with repetition allowed. So, total permutations are n^r = 5⁵ = 3125.

We know, ⁿC_r = \(\frac{n!}{(n-r)! r!}\).
Replacing r by n-r, we get ⁿC_n-r = \(\frac{n!}{(n-(n-r))!(n-r)!} = \frac{n!}{r! (n-r)!}\) = ⁿC_r
=> ⁿC_r = ⁿC_n-r

The number of permutations of n different objects taken r at a time, where repetition is allowed is n*n*n*n*n?????. r times = n^r.

1-9 digits which means 9 digits are possible. We have to arrange 3 digits at a time out of 9 digits with repetition allowed. So, total permutations are n^r = 9³ = 729.

Since re are 26 letters in English alphabet and 9 possible digits from 1 to 9.
So, by fundamental principle of counting, total possible codes are 26*9 = 234.

We know, ⁿP_r = \(\frac{n!}{(n-r)!}\).
ⁿP_n = \(\frac{n!}{(n-n)!} = \frac{n!}{(0)!}\) = n!

We know, ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r
Substituting n=14 and r=4, we get ¹⁴C_r + ¹⁴C_r-1 = ¹⁵C_r
=>¹⁴C_r-1 = ¹⁵C_r – ¹⁴C_r = 15-14 = 1.

PULSE is a 5 letters word. By fundamental principle of counting, total number of possible words with repetition allowed are 5*5*5*5 = 625.

Words starting with letter A comes first in dictionary. Starting with A, number of words = 4! = 24. Starting with E, number of words = 4! = 24. Starting with I, number of words = 4! = 24. Starting with G, number of words = 4! = 24. Since our word also start with M so, we have to consider one more letter i.e. MA. Since our word also start with MA so, we have to consider one more letter i.e. MAE. Starting with MAE, number of words = 2! = 2. Since our word also start with MAG so, we have to consider one more letter i.e. MAGE. Starting with MAGE, only one letter i.e. MAGEI. After this, MAGIE comes. Total number of words before MAGIE = 24 + 24 + 24 + 24 + 2 = 98. So, rank of MAGIE is 99.

If re are 4 paths to travel from Delhi to Kanpur n number of paths to travel from Kanpur to Delhi are 4. So, by fundamental principle of counting, total number of ways to go and come back are 4*4 = 16.