The ten's digit in 1! + 2! + 3! + ………..+49! Is __________.

Answer

To find the ten's digit in the sum \(1! + 2! + 3! + \ldots + 49!\), we can follow these steps:

### Step 1: Calculate Factorials from \(1!\) to \(10!\)
We start by calculating the factorials of the numbers from 1 to 10, as these will contribute to the sum significantly before the factorials grow large enough to end in zeros.

- \(1! = 1\)
- \(2! = 2\)
- \(3! = 6\)
- \(4! = 24\)
- \(5! = 120\)
- \(6! = 720\)
- \(7! = 5040\)
- \(8! = 40320\)
- \(9! = 362880\)
- \(10! = 3628800\)

### Step 2: Identify the Contribution of Factorials
From \(5!\) onward, the factorials will contribute increasingly larger numbers, and importantly, starting from \(5!\), every factorial will end with at least one zero. This is because \(5!\) includes the factor \(5\) and \(2\), which together form a \(10\).

### Step 3: Sum the Factorials
Now, we sum the factorials calculated above:

\[ 1! + 2! + 3! + 4! + 5! + 6! + 7! + 8! + 9! + 10! = 1 + 2 + 6 + 24 + 120 + 720 + 5040 + 40320 + 362880 + 3628800 \]

Calculating this step-by-step:

- \(1 + 2 = 3\)
- \(3 + 6 = 9\)
- \(9 + 24 = 33\)
- \(33 + 120 = 153\)
- \(153 + 720 = 873\)
- \(873 + 5040 = 5913\)
- \(5913 + 40320 = 46233\)
- \(46233 + 362880 = 409113\)
- \(409113 + 3628800 = 4037913\)

### Step 4: Determine the Ten's Digit
Now, we need to find the ten's digit of the final sum \(4037913\). The ten's digit is the second last digit in this number.

Looking at \(4037913\), the ten's digit is \(1\).

### Final Answer
Thus, the ten's digit in \(1! + 2! + 3! + \ldots + 49!\) is **1**.

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