Find the no. of zeros in experssion :
`1^(2) xx 2^(2) xx 3^(3) xx 4^(4) xx ……..xx100^(100)`

Options

A. 1200
B. 1300
C. 250
D. 1050

Answer

To find the number of zeros in the expression \(1^1 \times 2^2 \times 3^3 \times 4^4 \times \ldots \times 100^{100}\), we need to determine how many times the factor 10 appears in the product. Since \(10 = 2 \times 5\), we will find the number of factors of 2 and 5 in the expression and the limiting factor will determine the number of zeros.

### Step 1: Count the factors of 5

To find the number of factors of 5 in the expression, we will consider each term \(n^n\) for \(n = 1\) to \(100\). The number of factors of 5 in \(n^n\) is given by \(n \times \text{(number of factors of 5 in n)}\).

The number of factors of 5 in \(n\) can be calculated using the formula:
\[ \text{Number of factors of 5 in } n = \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \right\rfloor + \left\lfloor \frac{n}{125} \right\rfloor + \ldots \]

Let's calculate it for each \(n\) from 1 to 100.

- For \(n = 1\): \(0\)
- For \(n = 2\): \(0\)
- For \(n = 3\): \(0\)
- For \(n = 4\): \(0\)
- For \(n = 5\): \(1\)
- For \(n = 6\): \(1\)
- For \(n = 7\): \(1\)
- For \(n = 8\): \(1\)
- For \(n = 9\): \(1\)
- For \(n = 10\): \(2\)
- For \(n = 11\): \(2\)
- For \(n = 12\): \(2\)
- For \(n = 13\): \(2\)
- For \(n = 14\): \(2\)
- For \(n = 15\): \(3\)
- For \(n = 16\): \(3\)
- For \(n = 17\): \(3\)
- For \(n = 18\): \(3\)
- For \(n = 19\): \(3\)
- For \(n = 20\): \(4\)
- For \(n = 21\): \(4\)
- For \(n = 22\): \(4\)
- For \(n = 23\): \(4\)
- For \(n = 24\): \(4\)
- For \(n = 25\): \(6\)
- For \(n = 26\): \(6\)
- For \(n = 27\): \(6\)
- For \(n = 28\): \(6\)
- For \(n = 29\): \(6\)
- For \(n = 30\): \(7\)
- For \(n = 31\): \(7\)
- For \(n = 32\): \(7\)
- For \(n = 33\): \(7\)
- For \(n = 34\): \(7\)
- For \(n = 35\): \(8\)
- For \(n = 36\): \(8\)
- For \(n = 37\): \(8\)
- For \(n = 38\): \(8\)
- For \(n = 39\): \(8\)
- For \(n = 40\): \(9\)
- For \(n = 41\): \(9\)
- For \(n = 42\): \(9\)
- For \(n = 43\): \(9\)
- For \(n = 44\): \(9\)
- For \(n = 45\): \(10\)
- For \(n = 46\): \(10\)
- For \(n = 47\): \(10\)
- For \(n = 48\): \(10\)
- For \(n = 49\): \(12\)
- For \(n = 50\): \(12\)
- For \(n = 51\): \(12\)
- For \(n = 52\): \(12\)
- For \(n = 53\): \(12\)
- For \(n = 54\): \(12\)
- For \(n = 55\): \(13\)
- For \(n = 56\): \(13\)
- For \(n = 57\): \(13\)
- For \(n = 58\): \(13\)
- For \(n = 59\): \(13\)
- For \(n = 60\): \(14\)
- For \(n = 61\): \(14\)
- For \(n = 62\): \(14\)
- For \(n = 63\): \(14\)
- For \(n = 64\): \(14\)
- For \(n = 65\): \(15\)
- For \(n = 66\): \(15\)
- For \(n = 67\): \(15\)
- For \(n = 68\): \(15\)
- For \(n = 69\): \(15\)
- For \(n = 70\): \(16\)
- For \(n = 71\): \(16\)
- For \(n = 72\): \(16\)
- For \(n = 73\): \(16\)
- For \(n = 74\): \(16\)
- For \(n = 75\): \(17\)
- For \(n = 76\): \(17\)
- For \(n = 77\): \(17\)
- For \(n = 78\): \(17\)
- For \(n = 79\): \(17\)
- For \(n = 80\): \(18\)
- For \(n = 81\): \(18\)
- For \(n = 82\): \(18\)
- For \(n = 83\): \(18\)
- For \(n = 84\): \(18\)
- For \(n = 85\): \(19\)
- For \(n = 86\): \(19\)
- For \(n = 87\): \(19\)
- For \(n = 88\): \(19\)
- For \(n = 89\): \(19\)
- For \(n = 90\): \(20\)
- For \(n = 91\): \(20\)
- For \(n = 92\): \(20\)
- For \(n = 93\): \(20\)
- For \(n = 94\): \(20\)
- For \(n = 95\): \(21\)
- For \(n = 96\): \(21\)
- For \(n = 97\): \(21\)
- For \(n = 98\): \(21\)
- For \(n = 99\): \(21\)
- For \(n = 100\): \(24\)

Now, we sum these values for \(n = 1\) to \(100\):
\[ \text{Total factors of 5} = 0 + 0 + 0 + 0 + 1 + 1 + 1 + 1 + 1 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 3 + 4 + 4 + 4 + 4 + 4 + 6 + 6 + 6 + 6 + 6 + 7 + 7 + 7 + 7 + 7 + 8 + 8 + 8 + 8 + 8 + 9 + 9 + 9 + 9 + 9 + 10 + 10 + 10 + 10 + 12 + 12 + 12 + 12 + 12 + 12 + 13 + 13 + 13 + 13 + 13 + 14 + 14 + 14 + 14 + 14 + 15 + 15 + 15 + 15 + 15 + 16 + 16 + 16 + 16 + 16 + 17 + 17 + 17 + 17 + 17 + 18 + 18 + 18 + 18 + 18 + 19 + 19 + 19 + 19 + 19 + 20 + 20 + 20 + 20 + 20 + 21 + 21 + 21 + 21 + 21 + 24 = 124 \]

### Step 2: Count the factors of 2

Now we will count the factors of 2 in the same way. The number of factors of 2 in \(n^n\) is given by \(n \times \text{(number of factors of 2 in n)}\).

Using the same formula for counting factors of 2:
\[ \text{Number of factors of 2 in } n = \left\lfloor \frac{n}{2} \right\rfloor + \left\lfloor \frac{n}{4} \right\rfloor + \left\lfloor \frac{n}{8} \right\rfloor + \left\lfloor \frac{n}{16} \right\rfloor + \left\lfloor \frac{n}{32} \right\rfloor + \left\lfloor \frac{n}{64} \right\rfloor + \ldots \]

Calculating for \(n = 1\) to \(100\):
- For \(n = 1\): \(0\)
- For \(n = 2\): \(1\)
- For \(n = 3\): \(1\)
- For \(n = 4\): \(3\)
- For \(n = 5\): \(3\)
- For \(n = 6\): \(4\)
- For \(n = 7\): \(4\)
- For \(n = 8\): \(7\)
- For \(n = 9\): \(7\)
- For \(n = 10\): \(8\)
- For \(n = 11\): \(8\)
- For \(n = 12\): \(10\)
- For \(n = 13\): \(10\)
- For \(n = 14\): \(11\)
- For \(n = 15\): \(11\)
- For \(n = 16\): \(15\)
- For \(n = 17\): \(15\)
- For \(n = 18\): \(17\)
- For \(n = 19\): \(17\)
- For \(n = 20\): \(22\)
- For \(n = 21\): \(22\)
- For \(n = 22\): \(24\)
- For \(n = 23\): \(24\)
- For \(n = 24\): \(29\)
- For \(n = 25\): \(29\)
- For \(n = 26\): \(31\)
- For \(n = 27\): \(31\)
- For \(n = 28\): \(36\)
- For \(n = 29\): \(36\)
- For \(n = 30\): \(41\)
- For \(n = 31\): \(41\)
- For \(n = 32\): \(47\)
- For \(n = 33\): \(47\)
- For \(n = 34\): \(52\)
- For \(n = 35\): \(52\)
- For \(n = 36\): \(59\)
- For \(n = 37\): \(59\)
- For \(n = 38\): \(64\)
- For \(n = 39\): \(64\)
- For \(n = 40\): \(71\)
- For \(n = 41\): \(71\)
- For \(n = 42\): \(78\)
- For \(n = 43\): \(78\)
- For \(n = 44\): \(85\)
- For \(n = 45\): \(85\)
- For \(n = 46\): \(92\)
- For \(n = 47\): \(92\)
- For \(n = 48\): \(101\)
- For \(n = 49\): \(101\)
- For \(n = 50\): \(110\)
- For \(n = 51\): \(110\)
- For \(n = 52\): \(119\)
- For \(n = 53\): \(119\)
- For \(n = 54\): \(128\)
- For \(n = 55\): \(128\)
- For \(n = 56\): \(137\)
- For \(n = 57\): \(137\)
- For \(n = 58\): \(146\)
- For \(n = 59\): \(146\)
- For \(n = 60\): \(156\)
- For \(n = 61\): \(156\)
- For \(n = 62\): \(165\)
- For \(n = 63\): \(165\)
- For \(n = 64\): \(175\)
- For \(n = 65\): \(175\)
- For \(n = 66\): \(184\)
- For \(n = 67\): \(184\)
- For \(n = 68\): \(193\)
- For \(n = 69\): \(193\)
- For \(n = 70\): \(203\)
- For \(n = 71\): \(203\)
- For \(n = 72\): \(213\)
- For \(n = 73\): \(213\)
- For \(n = 74\): \(223\)
- For \(n = 75\): \(223\)
- For \(n = 76\): \(233\)
- For \(n = 77\): \(233\)
- For \(n = 78\): \(243\)
- For \(n = 79\): \(243\)
- For \(n = 80\): \(253\)
- For \(n = 81\): \(253\)
- For \(n = 82\): \(263\)
- For \(n = 83\): \(263\)
- For \(n = 84\): \(273\)
- For \(n = 85\): \(273\)
- For \(n = 86\): \(283\)
- For \(n = 87\): \(283\)
- For \(n = 88\): \(293\)
- For \(n = 89\): \(293\)
- For \(n = 90\): \(303\)
- For \(n = 91\): \(303\)
- For \(n = 92\): \(313\)
- For \(n = 93\): \(313\)
- For \(n = 94\): \(323\)
- For \(n = 95\): \(323\)
- For \(n = 96\): \(333\)
- For \(n = 97\): \(333\)
- For \(n = 98\): \(343\)
- For \(n = 99\): \(343\)
- For \(n = 100\): \(353\)

Now, we sum these values for \(n = 1\) to \(100\):
\[ \text{Total factors of 2} = 0 + 1 + 1 + 3 + 3 + 4 + 4 + 7 + 7 + 8 + 8 + 10 + 10 + 11 + 11 + 15 + 15 + 17 + 17 + 22 + 22 + 24 + 24 + 29 + 29 + 31 + 31 + 36 + 36 + 41 + 41 + 47 + 47 + 52 + 52 + 59 + 59 + 64 + 64 + 71 + 71 + 78 + 78 + 85 + 85 + 92 + 92 + 101 + 101 + 110 + 110 + 119 + 119 + 128 + 128 + 137 + 137 + 146 + 146 + 156 + 156 + 165 + 165 + 175 + 175 + 184 + 184 + 193 + 193 + 203 + 203 + 213 + 213 + 223 + 223 + 233 + 233 + 243 + 243 + 253 + 253 + 263 + 263 + 273 + 273 + 283 + 283 + 293 + 293 + 303 + 303 + 313 + 313 + 323 + 323 + 333 + 333 + 343 + 343 + 353 = 1287 \]

### Step 3: Determine the number of zeros

The number of zeros at the end of the product is determined by the minimum of the total factors of 2 and the total factors of 5:
\[ \text{Number of zeros} = \min(1287, 124) = 124 \]

Thus, the number of zeros in the expression \(1^1 \times 2^2 \times 3^3 \times 4^4 \times \ldots \times 100^{100}\) is **124**.